Bounding Regions to Plane Steepest Descent Curves of Quasi Convex families

Research Article

Bounding Regions to Plane Steepest Descent Curves of

Quasiconvex Families

Marco Longinetti,

Paolo Manselli,

and Adriana Venturi

1_{Dipartimento di Matematica e Informatica Ulisse Dini, Universit`a degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy</sub> 2<sub>Dipartimento GESAAF, Universit`a degli Studi di Firenze, Piazzale delle Cascine 15, 50144 Firenze, Italy}

Correspondence should be addressed to Marco Longinetti; marco.longinetti@unifi.it Received 26 February 2016; Accepted 17 April 2016

Academic Editor: Wenyu Sun

Copyright © 2016 Marco Longinetti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with

constraints) outside of a convex body𝐾 is studied. It is shown that possible extensions are constrained to lie inside of suitable

bounding regions depending on𝐾. These regions are bounded by arcs of involutes of 𝜕𝐾 and satisfy many inclusions properties.

The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.

1. Introduction

Let𝑢 be a smooth function defined in a convex body Ω ⊂ R𝑛.

Let𝐷𝑢(𝑥) ̸= 0 in {𝑥 ∈ Ω : 𝑢(𝑥) > min 𝑢}. A classical steepest

descent curve of𝑢 is a rectifiable curve 𝑠 → 𝑥(𝑠) solution to

𝑑𝑥

𝑑𝑠 = −

𝐷𝑢

|𝐷𝑢|(𝑥 (𝑠)) . (1)

Classical steepest descent curves are the integral curves of a unit field normal to the sublevel sets of the given smooth

function𝑢. We are interested in “generalized” steepest descent

curves that are integral curves to a unit field normal to a

nested family of convex sets {Ω_𝑡} (see Definition 5); {Ω_𝑡}

will be called a quasiconvex family as in [1]. Sharp bounds about the length of the steepest descent curves for a quasi convex family have been proved in [2–4]. The geometry of these curves, equivalent definitions, related questions and generalizations have been studied in [5–9].

In the present work generalized steepest descent curves for

a quasiconvex family (SDC for short) are defined as bounded

oriented rectifiable curves𝛾 ⊂ R𝑛, with a locally Lipschitz

continuous parameterization 𝑇 ∋ 𝑡 → 𝑥(𝑡), with ascent

parameter, satisfying

⟨ ̇𝑥 (𝑡) , 𝑥 (𝜏) − 𝑥 (𝑡)⟩ ≤ 0, a.e. 𝑡 ∈ 𝑇, ∀𝜏 ≤ 𝑡; (2)

⟨⋅, ⋅⟩ is the scalar product in R𝑛. Let ordering⪯ be chosen on

𝛾, according to the orientation; let us denote

𝛾𝑥= {𝑦 ∈ 𝛾 : 𝑦 ⪯ 𝑥} . (3)

In [8, Theorem 4.10], the SDC are characterized in an equiv-alent way as self-distancing curves, namely, oriented (⪯)

continuous curves with the property that the distance of𝑥

to an arbitrarily fixed previous point𝑥₁is not decreasing:

𝑥₁⪯ 𝑥₂⪯ 𝑥₃󳨐⇒

󵄨󵄨󵄨󵄨𝑥2− 𝑥1󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑥3− 𝑥1󵄨󵄨󵄨󵄨 ∀𝑥1, 𝑥2, 𝑥3∈ 𝛾.

(4) Thus steepest descent curves are self-distancing curves and both denoted SDC. In [8] self-distancing curves are called self-expanding curves. With the opposite orientation these curves have been also introduced, studied, and called self-approaching curves (see [2]) or self-contracting curves (see [7]).

Volume 2016, Article ID 4873276, 17 pages http://dx.doi.org/10.1155/2016/4873276

In our work we are interested in the behaviour and

properties of a plane SDC𝛾 beyond its final point 𝑥₀. One of

the principal goals of the paper is to show that conditions (2)

and (4) imply constraints for possible extensions of curve𝛾

beyond𝑥₀; these constraints are written as bounding regions

for the possible extensions of𝛾_𝑥0.

An important property that will be used later is the

property of distancing from a set𝐴.

Definition 1. Given a set𝐴, an absolutely continuous curve 𝛾,

𝑇 ∋ 𝑡 → 𝑥(𝑡) has the distancing from 𝐴 property if it satisfies

⟨ ̇𝑥 (𝑡) , 𝑦 − 𝑥 (𝑡)⟩ ≤ 0, a.e. 𝑡 ∈ 𝑇, ∀𝑦 ∈ 𝐴 . (5)

Let us outline the content of our work. In Section 2 introductory definitions are given and covering maps for the boundary of a plane convex set, needed for later use, are introduced. In Section 3 the involutes of the boundary of a plane convex body are introduced and some of their properties are proved.

In Section 4 plane regions depending on the convex hull

of𝛾_𝑥0have been defined; these regions fence in or fence out

the possible extensions of 𝛾_𝑥0. The boundary of these sets

consists of arcs of involutes of convex bodies, constructed in Section 3. As an application, in Section 4.1 the following

problem has been studied: given a convex set𝐾, 𝑥₀∈ 𝜕𝐾, 𝑥₁∉

𝐾, is it possible to construct SDC joining 𝑥₀to𝑥₁, satisfying

the distancing from K property? Minimal properties of this construction have been introduced and studied. In Section 5

sets of points more general than SDC are studied. A set𝜎 ⊂

R2_{(not necessarily a curve) of ordered points satisfying (4)}

will be called self-distancing set; see also Definition 2; with

the opposite order,𝜎 was called in [6] self-contracting set

and many properties of these sets, as only subsets of self-contracting curves, were there obtained. Another goal of the paper is the solution to the following question: given a

self-distancing set𝜎 ⊂ R2 does a steepest descent curve𝛾 ⊃

𝜎 exist? In Section 5 examples, necessary and/or sufficient

conditions are given when𝜎 consists of a finite or countable

number of points 𝑥_𝑖 ∈ R2 and/or steepest descent curves

𝛾𝑖_{⊂ R}2_.

In the present work the two-dimensional case is studied.

Similar results for the 𝑛-dimensional case are an open

problem stated at the end of the work.

2. Preliminaries and Definitions

Let

𝐵 (𝑧, 𝜌) = {𝑥 ∈ R𝑛_{: |𝑥 − 𝑧| < 𝜌} ,}

𝑆𝑛−1= 𝜕𝐵 (0, 1)

𝑛 ≥ 2.

(6)

A nonempty, compact convex set𝐾 of R𝑛 will be called a

convex body. From now on,𝐾 will always be a convex body not reduced to a point. Int(𝐾) and 𝜕𝐾 denote the interior of 𝐾 and

the boundary of𝐾, |𝜕𝐾| denotes its length, cl(𝐾) is the closure

of𝐾, Aff(𝐾) will be the smallest affine space containing 𝐾,

and relint𝐾 and 𝜕_rel𝐾 are the corresponding subsets in the

topology of Aff(𝐾). For every set 𝑆 ⊂ R𝑛, co(𝑆) is the convex

hull of𝑆.

Let𝑞 ∈ 𝐾; the normal cone at 𝑞 to 𝐾 is the closed convex

cone:

𝑁_𝐾(𝑞) = {𝑥 ∈ R𝑛: ⟨𝑥, 𝑦 − 𝑞⟩ ≤ 0 ∀𝑦 ∈ 𝐾} . (7)

When𝑞 ∈ Int(𝐾), then 𝑁_𝐾(𝑞) reduces to zero.

The tangent cone or support cone of K at a point𝑞 ∈ 𝜕𝐾

is given by

𝑇_𝐾(𝑞) = cl ( ⋃

𝑦∈𝐾

{𝑠 (𝑦 − 𝑞) : 𝑠 ≥ 0}) . (8)

In two dimensions cones will be called sectors.

Let𝐾 be a convex body and let 𝑝 be a point. A simple cap

body𝐾𝑝is

𝐾𝑝= ⋃

0≤𝜆≤1{𝜆𝐾 + (1 − 𝜆) 𝑝} = co (𝐾 ∪ {𝑝}) . (9)

Cap bodies properties can be found in [10, 11].

2.1. Self-Distancing Sets and Steepest Descent Curves. Let us

recall the following definitions.

Definition 2. Let us call self-distancing set a bounded subset𝜎

ofR𝑛, linearly ordered (by⪯), with the property

𝑥₁⪯ 𝑥₂⪯ 𝑥₃󳨐⇒

󵄨󵄨󵄨󵄨𝑥2− 𝑥1󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑥3− 𝑥1󵄨󵄨󵄨󵄨, 𝑥1, 𝑥2, 𝑥3∈ 𝜎.

(10) The self-distancing sets have been introduced in [6] with

the opposite order. If a self-distancing set 𝜎 is a closed

connected set, not reduced to a point, then it can be proved

that 𝜎 is the support of a steepest descent curve 𝛾 (see [8,

Theorem 4.10, Theorem 4.8]) and it will also be called a

self-distancing curve𝛾.

The short name SDC will be used both for self-distancing curves and for steepest descent curves in all the paper.

Definition 3. Let𝐾 be a convex body; 𝛾 ⊂ R2\ relint 𝐾 will

be called a self-distancing curve from𝐾 (denoted SDC_𝐾) if

(i)𝛾 is a self-distancing curve,

(ii)𝛾 ∩ 𝜕_rel𝐾 ̸= 0,

(iii)𝛾 has the property

𝑥 ⪯ 𝑥₁󳨐⇒

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑥1− 𝑦󵄨󵄨󵄨󵄨 ∀𝑦 ∈ 𝐾, ∀𝑥, 𝑥1∈ 𝛾.

(11)

When (ii) does not hold, that is𝛾 ∩ 𝜕_rel𝐾 = 0, 𝛾 will be called

a deleted self-distancing curve from𝐾.

Remark 4. Let𝛾 be SDC_𝐾, since𝛾 has an absolutely continu-ous parameterization ([8, Theorem 4.10, Theorem 4.8]), thus

If𝛾 is SDC and 𝑥 ∈ 𝛾 then 𝛾 \ 𝛾_𝑥 is SDC_co(𝛾𝑥). That is,

the tangent vector ̇𝑥(𝑡) to 𝛾 is in the normal cone at 𝑥(𝑡) to

the related convex setΩ_𝑡 fl co(𝛾_𝑥(𝑡)). This condition is a

generalization of the classicals steepest descent curves that are integral curves to a unit field normal to smooth quasiconvex families.

Nested families of convex sets have been introduced and studied by de Finetti [12] and Fenchel [1]. Let us recall some definitions.

Definition 5. Let𝑇 be a real interval. A convex stratification

(see [12]) is a nonempty familyK of convex bodies Ω_𝑡⊂ R𝑛,

𝑡 ∈ 𝑇 ⊂ R, linearly strictly ordered by inclusion (Ω₁ ⊂ Ω₂,

Ω₁ ̸= Ω₂), with a maximum set (maxK) and a minimum set

(minK).

LetK = {Ω_𝑡}_𝑡∈𝑇be a convex stratification. If for every

𝑠 ∈ 𝑇 \ {max 𝑇} the property ⋂

𝑡>𝑠 Ω𝑡= Ω𝑠 (12)

holds, then as in [1],K = {Ω_𝑡}_𝑡∈𝑇will be called a quasiconvex

family.

An important quasiconvex family associated with a

con-tinuous self-distancing curve from𝐾, 𝛾: 𝑡 → 𝑥(𝑡) is K =

{Ω_𝑡}_𝑡∈𝑇, where

Ω_𝑡= co (𝛾_𝑥(𝑡)∪ 𝐾) . (13)

The couple(𝛾, K) is special case of Expanding Couple, a class

introduced in [8].

Remark 6. If𝛾 ∈ SDC_𝐾, then for all𝑥 ∈ 𝛾 the curve (𝛾 \ 𝛾_𝑥) ∪ {𝑥} is a self-distancing curve from the convex hull of the set

𝛾_𝑥∪ 𝐾.

This fact is a direct consequence of the following.

Proposition 7 (see [8, Lemma 4.9]). Let 𝑝, 𝑞, 𝑦_𝑖 ∈ R𝑛_{,𝑖 =}

1, . . . , 𝑠. If

󵄨󵄨󵄨󵄨𝑝 − 𝑦󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑞 − 𝑦󵄨󵄨󵄨󵄨, 𝑓𝑜𝑟 𝑦 = 𝑦𝑖, 𝑖 = 1, . . . , 𝑠 (14)

then the same holds for every𝑦 ∈ co({𝑦_𝑖, 𝑖 = 1, . . . , 𝑠}).

The statement of the previous proposition holds if large inequalities are replaced by strict inequalities everywhere.

2.2. The Support Function of a Plane Convex Body. Let𝐾 ⊂

R𝑛_{be a convex body not reduced to a point.}

For a convex body𝐾, the support function is defined as

𝐻𝐾(𝑥) = sup

𝑦∈𝐾⟨𝑥, 𝑦⟩ , 𝑥 ∈ R

𝑛_,

(15)

where⟨⋅, ⋅⟩ denotes the scalar product in R𝑛. For𝑛 = 2, 𝜗 ∈ R,

let𝜃 = (cos 𝜗, sin 𝜗) ∈ 𝑆1 andℎ_𝐾(𝜗) fl 𝐻_𝐾(𝜃); it will be

denotedℎ(𝜗) if no ambiguity arises.

For every𝜃 ∈ 𝑆1there exists at least one point𝑥 ∈ 𝜕𝐾

such that

⟨𝜃, 𝑦 − 𝑥⟩ ≤ 0 ∀𝑦 ∈ 𝐾; (16)

this means that the line through𝑥 orthogonal to 𝜃 supports

𝐾. For every 𝑥 ∈ 𝜕𝐾 let ̂𝑁𝑥be the set of𝜃 ∈ 𝑆1such that (16)

holds. Let𝐹(𝜃) be the set of all 𝑥 ∈ 𝜕𝐾 satisfying (16). If 𝜕𝐾

is strictly convex at the direction𝜃 then 𝐹(𝜃) reduces to one

point and it will be denoted by𝑥(𝜃).

Definition 8. The set valued map𝐺 : 𝜕𝐾 → 𝑆1, 𝜕𝐾 ∋ 𝑥 →

̂

𝑁_𝑥 ⊂ 𝑆1_{, is the generalized Gauss map;𝑥 ∈ 𝜕𝐾 is a vertex on}

𝜕𝐾 iff ̂𝑁_𝑥is a sector with interior points. The set valued map

𝐹 : 𝑆1 _{→ 𝜕𝐾, 𝑆}1 _{∋ 𝜃 → 𝐹(𝜃) ⊂ 𝜕𝐾 is the reverse generalized}

Gauss map;𝐹(𝜃) is a closed segment, possibly reduced to a

single point, and it will be called 1-face when it has interior points.

Let𝑃 be the covering map

𝑃 : R 󳨀→ 𝑆1, (17)

R ∋ 𝜗 󳨀→

𝜃 = (cos 𝜗, sin 𝜗) ∈ 𝑆1. (18)

Let𝐿 = |𝜕𝐾|, and let 𝑠 → 𝑥_𝑙(𝑠), 0 ≤ 𝑠 < 𝐿 (𝑠 → 𝑥_𝑟(𝑠),

0 ≤ 𝑠 < 𝐿) be the parametric representations of 𝜕𝐾 depending on the arc length counterclockwise (clockwise) with an initial

point (not necessarily the same). Let us extend𝑥_𝑙(⋅) and 𝑥_𝑟(⋅)

by defining

𝑥_𝑙(𝑠) fl 𝑥_𝑙(𝑠 − 𝑘𝐿) if 𝑘𝐿 ≤ 𝑠 < (𝑘 + 1) 𝐿, (𝑘 ∈ Z) , (19)

similarly for𝑥_𝑟.

Let us fix𝑥₀ ∈ 𝜕𝐾, 𝜃₀ ∈ 𝐺(𝑥₀), 𝜃₀ = (cos 𝜗₀, sin 𝜗₀), and

𝜗₀∈ R.

For later use, we need to have 𝑥₀ = 𝑥_𝑙(𝑠₀) = 𝑥_𝑟(𝑠₀);

this can be realized by choosing suitable initial points for the

parameterizations𝑥_𝑙and𝑥_𝑟. Then 𝑥_𝑙(𝑠₀+ 𝑠) = 𝑥_𝑟(𝑠₀+ 𝐿 − 𝑠) , ∀𝑠 ∈ R. (20) The maps 𝑥_𝑙: R 󳨀→ 𝜕𝐾, 𝑥_𝑟: R 󳨀→ 𝜕𝐾 (21)

are covering maps.

The initial parameters will be

𝑥₀= 𝑥_𝑙(𝑠₀) = 𝑥_𝑟(𝑠₀) ∈ 𝜕𝐾,

𝑆1_{∋ 𝜃}

0∈ 𝐹−1(𝑥0) ,

R ∋ 𝜗₀∈ 𝑃−1(𝜃₀)

(22)

(𝐹−1(𝑥0), 𝑃−1(𝜃0) are the back images of 𝐹, 𝑃, resp.). Let 𝑘 ∈

Z. Let us define, for 𝜗0+ 2𝑘𝜋 < 𝜗 < 𝜗0+ 2(𝑘 + 1)𝜋:

𝑠_𝑙+(𝜗) fl sup {𝑠 ∈ R : 𝑘𝐿 < 𝑠 ≤ (𝑘 + 1) 𝐿, 𝑥_𝑙(𝑠)

if𝜗 = 𝜗₀+ 2𝑘𝜋

𝑠_𝑙+(𝜗) fl sup {𝑠 ∈ R : 𝑘𝐿 ≤ 𝑠 < (𝑘 + 1) 𝐿, 𝑥_𝑙(𝑠)

∈ 𝐹 (𝑃 (𝜗))} . (24)

Similarly, let us define for𝜗₀+ 2(𝑘 − 1)𝜋 < 𝜗 < 𝜗₀+ 2𝑘𝜋:

𝑠_𝑟−(𝜗) fl inf {𝑠 ∈ R : 𝑘𝐿 ≤ 𝑠 < (𝑘 + 1) 𝐿, 𝑥_𝑟(𝑠)

∈ 𝐹 (𝑃 (𝜗))} ; (25)

if𝜗 = 𝜗₀+ 2𝑘𝜋

𝑠_𝑟−(𝜗) fl inf {𝑠 ∈ R : (𝑘 − 1) 𝐿 < 𝑠 ≤ 𝑘𝐿, 𝑥_𝑟(𝑠)

∈ 𝐹 (𝑃 (𝜗))} . (26)

The function𝑠_𝑙+is increasing inR and right continuous and

with left limits (so called cadlag function). Similar properties

hold for−𝑠_𝑟−. Let us recall that a cadlag increasing function

𝑠(𝜗), 𝜗 ∈ R, has a right continuous inverse defined as

𝜗 (𝑠) = inf {𝜗 : 𝑠 (𝜗) > 𝑠} . (27)

Let𝜗_𝑙+(⋅) be the right continuous inverse of 𝑠_𝑙+(⋅). Let 𝑠 →

𝜗_𝑟−(𝑠) be the opposite of the right continuous inverse of

−𝑠_𝑟−(⋅).

Let us introduce for simplicity

n_𝜗fl (cos 𝜗, sin 𝜗) ,

t_𝜗fl (− sin 𝜗, cos 𝜗) . (28)

Let𝜗 → ℎ(𝜗) be the support function of 𝐾.

It is well known ([13]) that if𝜕𝐾 is 𝐶2₊(i.e.,𝜕𝐾 ∈ 𝐶2, with

positive curvature), thenℎ is 𝐶2 and the counterclockwise

element arc𝑑𝑠 of 𝜕𝐾 is given by

𝑑𝑠 = (ℎ + ̈ℎ) 𝑑𝜗. (29)

ℎ(𝜗) + ̈ℎ(𝜗) is the positive radius of curvature; moreover the

reverse Gauss map𝐹 : 𝜃 → 𝑥 ∈ 𝜕𝐾 is a 1-1 map given by

𝑥 (𝜃) fl ℎ (𝜗) n𝜗+ ̇ℎ (𝜗) t𝜗, 𝜗 ∈ 𝑃−1(𝜃) . (30)

The previous formula also holds for an arbitrary convex

body, for every𝜗 such that 𝐹(𝜃) is reduced to a point; see [10].

Let us recall that a real valued function𝑥 → 𝑓(𝑥) is called

semiconvex onR when there exists a positive constant 𝐶 such

that𝑓(𝑥) + 𝐶𝑥2is convex onR. From (29) the function 𝜗 →

ℎ(𝜗) + (1/2)𝜗2maxℎ is convex on R; thus ℎ is semiconvex. In

the case that𝐾 is an arbitrary convex body, by approximation

arguments with𝐶2₊convex bodies (see [11]) it follows that the

support function of𝐾 is also semiconvex. As consequence ℎ

is Lipschitz continuous, it has left (right) derivative ̇ℎ₋(resp.,

̇ℎ₊) at each point, which is left (right) continuous. Moreover

at each point the right limit of ̇ℎ₋is ̇ℎ₊and the left limit of ̇ℎ₊

is ̇ℎ₋; see [14, pp. 228].

It is not difficult to show (from (30), with a right limit

argument) that, for an arbitrary convex body, for𝜗 ∈ R, the

formula

𝑥_𝑙(𝑠_𝑙+(𝜗)) = ℎ (𝜗) n_𝜗+ ̇ℎ₊(𝜗) t_𝜗 (31)

holds. Similarly the formula

𝑥_𝑟(𝑠_𝑟−(𝜗)) = ℎ (𝜗) n_𝜗+ ̇ℎ₋(𝜗) t_𝜗 (32)

holds.

If𝜕𝐾 is not strictly convex at the direction 𝜃 = (cos 𝜗,

sin𝜗) then ℎ is not differentiable at 𝜗 and

̇ℎ

+(𝜗) − ̇ℎ−(𝜗) = 󵄨󵄨󵄨󵄨𝑥𝑙(𝑠𝑙+(𝜗)) − 𝑥𝑟(𝑠𝑟−(𝜗))󵄨󵄨󵄨󵄨 = |𝐹 (𝜃)| . (33)

If𝑥₁, 𝑥₂ ∈ 𝜕𝐾 let us define arc+(𝑥₁, 𝑥₂) as the set of points

of𝜕𝐾 between 𝑥₁and𝑥₂according to the counterclockwise

orientation of 𝜕𝐾 and arc−(𝑥₁, 𝑥₂) as the set of points

between𝑥₁ and𝑥₂, according to the clockwise orientation;

|arc±_(𝑥

1, 𝑥2)| denote their length.

Remark 9. It is well known that a sequence of convex body

𝐾(𝑛)_{converges uniformly to𝐾 if and only if the}

correspond-ing sequence of support functions converges in the uniform norm; see [11, pp. 66]. Moreover as the two sequences of the end points of a closed counterclockwise oriented arc of

𝜕𝐾(𝑛)converge, then the sequence of the corresponding arcs

converges to a connected arc of𝜕𝐾 and the sequence of the

corresponding lengths converges too.

Proposition 10. Let 𝐾 be a convex body and ℎ its support

function; then 𝑠_𝑙+(𝜗) − 𝑠𝑙+(𝜗0) = ∫ 𝜗 𝜗0 ℎ (𝜏) 𝑑𝜏 + ( ̇ℎ+(𝜗) − ̇ℎ+(𝜗0)) , ∀𝜗 ≥ 𝜗₀; (34) 𝑠_𝑟−(𝜗₀) − 𝑠_𝑟−(𝜗) = ∫𝜗 𝜗0 ℎ (𝜏) 𝑑𝜏 + ( ̇ℎ−(𝜗) − ̇ℎ−(𝜗0)) , ∀𝜗 ≤ 𝜗₀. (35)

Proof. For every convex body𝐾 not reduced to a point the

function𝜗 → 𝑠_𝑙+(𝜗) is defined everywhere and satisfies the

weak form of (29); namely, − ∫

R𝑠𝑙+(𝜂) ̇𝜙 (𝜂) 𝑑𝜂 = ∫R(𝜙 + ̈𝜙) (𝜂) ℎ (𝜂) 𝑑𝜂,

∀𝜙 ∈ 𝐶∞₀ (R) .

(36)

Using the fact that 𝜗 → ℎ(𝜗) is Lipschitz continuous,

integrating by parts (36), the formula − ∫ R𝑠𝑙+(𝜂) ̇𝜙 (𝜂) 𝑑𝜂 = − ∫ R ̇𝜙 (𝜂) (∫ 𝜂 0 ℎ (𝜏) 𝑑𝜏 + ̇ℎ (𝜂)) 𝑑𝜂, ∀𝜙 ∈ 𝐶∞₀ (R) (37)

holds. Thus

𝑠_𝑙+(𝜂) = 𝑐 + ∫𝜂

0 ℎ (𝜏) 𝑑𝜏 + ̇ℎ (𝜂) , a.e. (38)

with𝑐 constant. Passing to the right limit, the equality

𝑠_𝑙+(𝜂) = 𝑐 + ∫𝜂

0 ℎ (𝜏) 𝑑𝜏 + ̇ℎ+(𝜂) , ∀𝜂 ∈ R (39)

holds. Formula (34) follows, by computing𝑠_𝑙+(𝜗) − 𝑠_𝑙+(𝜗₀),

using the previous equality. Similarly (35) is proved.

3. Involutes of a Closed Convex Curve

Definition 11. Let𝐼 be an interval. A plane curve 𝐼 ∋ 𝑡 → 𝑥(𝑡)

is convex if at every point𝑥 it has right tangent vector 𝑇+(𝑥)

and arg𝑇+(𝑥(𝑡)) is not decreasing function.

Let 𝑠 → 𝑥(𝑠) be the arc length parameterization of a

smooth curve; the classical definition of involute starting at

a point𝑥₀= 𝑥(𝑠₀) of the curve 𝑥(⋅) is

𝑖 (𝑠) = 𝑥 (𝑠) − (𝑠 − 𝑠0) 𝑥󸀠(𝑠) 𝑠 ≥ 𝑠0. (40)

Let us notice that𝑠 is the arc length of the curve, not of

the involute; if𝑠₀ = 0, then the starting point of the involute

coincides with the starting point of the curve. It is easy to construct an involute of a convex polygonal line (even if classical definition (40) does not work) by using arcs of circle centered at its corner points; moreover the involute depends on the orientation of the curve.

In this section, involutes for the boundary of an arbitrary

plane convex body𝐾, not reduced to a point, will be defined.

The assumption that𝐾 is an arbitrary convex body is needed

to work with the involutes of the convex sets, not smooth, studied in Section 4.

Let𝐾 ∈ 𝐶2₊; let𝑥₀be a fixed point of𝜕𝐾; 𝑠 → 𝑥(𝑠) can be

the clockwise parameterization of𝜕𝐾 or the counterclockwise

parameterization. Since there exist two orientations, then two different involutes have to be considered. As noted previously

one can assume that the parameterizations of𝜕𝐾 have been

chosen so that𝑥₀= 𝑥_𝑙(𝑠₀) = 𝑥_𝑟(𝑠₀).

Definition 12. Let one denote by 𝑖_𝑙,𝑥0 the left involute of

𝜕𝐾 starting at 𝑥₀ corresponding to the counterclockwise

parameterization of𝜕𝐾 and by 𝑖_𝑟,𝑥0the right involute

corre-sponding to the clockwise parameterization. When one needs

to emphasize the dependence on𝐾 of involutes, they will be

written as𝑖𝐾_𝑙,𝑥

0,𝑖

𝐾 𝑟,𝑥0.

Remark 13. Let us notice that if𝜌 is a plane reflection with

respect to a fixed axis then

𝑖𝐾_𝑟,𝑥0= 𝜌 (𝑖𝜌(𝐾)_{𝑙,𝜌(𝑥}

0)) . (41)

This relation allows us to prove our results for the left invo-lutes only and to state without proof the analogous results for the right involutes.

Theorem 14. Let one fix the initial parameters 𝑥₀,𝑠₀,𝜃₀, and

𝜗₀. The left and the right involutes of a plane convex curve starting at𝑥₀ ∈ 𝜕𝐾, boundary of a 𝐶2₊ plane convex body

𝐾 with support function ℎ, are parameterized by the value 𝜗

related to the outer normaln_𝜗to𝐾, as follows:

𝑖_𝑙,𝑥0(𝜗) = ℎ (𝜗) n𝜗− (∫ 𝜗 𝜗0 ℎ (𝜏) 𝑑𝜏 − ̇ℎ (𝜗0)) t𝜗, 𝑓𝑜𝑟 𝜗 ≥ 𝜗₀, (42) 𝑖𝑟,𝑥0(𝜗) = ℎ (𝜗) n𝜗− (∫ 𝜗 𝜗0 ℎ (𝜏) 𝑑𝜏 − ̇ℎ (𝜗0)) t𝜗, 𝑓𝑜𝑟 𝜗 ≤ 𝜗₀. (43)

Proof. In the present case there is a 1-1 mapping between𝜗

and𝑠; from (29), it follows that

𝑠 − 𝑠₀= ∫𝜗

𝜗0

ℎ (𝜏) 𝑑𝜏 + ̇ℎ (𝜗) − ̇ℎ (𝜗0) ; (44)

then, changing the variable𝑠 with 𝜗 in (40), with elementary

computation, (42) is obtained (since𝑥󸀠(𝑠) = t_𝜗 and (30)

holds). Formula (43) follows from (32) and (35).

For an arbitrary convex body𝐾 in place of (30), formulas

(31) and (32) have to be used.

Definition 15. Let𝐾 be a plane convex body; let

𝑥0= 𝑥 (𝑠0) ∈ 𝜕𝐾,

𝜗+₀ fl 𝜗_𝑙+(𝑠₀) ,

𝑠+₀ fl 𝑠𝑙+(𝜗+0) .

(45)

The left involute of𝜕𝐾 starting at 𝑥₀will be defined as

𝑖_𝑙,𝑥0(𝜗) = 𝑥𝑙(𝑠𝑙+(𝜗)) − (𝑠𝑙+(𝜗) − 𝑠0) t𝜗 for𝜗 ≥ 𝜗+0; (46)

similarly if𝜗−₀ := 𝜗_𝑟−(𝑠₀), 𝑠−₀ := 𝑠_𝑟−(𝜗−₀), the right involute

starting at𝑥₀will be defined as

𝑖_𝑟,𝑥0(𝜗) = 𝑥_𝑟(𝑠_𝑟−(𝜗)) + (𝑠_𝑟−(𝜗) − 𝑠₀) t_𝜗 for𝜗 ≤ 𝜗₀−. (47)

From (46) and (34) it follows that

𝑖_𝑙,𝑥0(𝜗) = ℎ (𝜗) n𝜗− (∫ 𝜗 𝜗+ 0 ℎ (𝜏) 𝑑𝜏 − ̇ℎ+(𝜗+0)) t𝜗 − 󵄨󵄨󵄨󵄨𝑥0− 𝑥𝑙(𝑠+0)󵄨󵄨󵄨󵄨 t𝜗, 𝜗 ≥ 𝜗+0; (48)

similarly from (47), (35) it follows that

𝑖_𝑟,𝑥0(𝜗) = ℎ (𝜗) n_𝜗− (∫𝜗 𝜗− 0 ℎ (𝜏) 𝑑𝜏 − ̇ℎ−(𝜗−0)) t𝜗 + 󵄨󵄨󵄨󵄨𝑥0− 𝑥𝑟(𝑠−0)󵄨󵄨󵄨󵄨 t𝜗, 𝜗 ≤ 𝜗0−. (49)

Let us notice that in (48) and (49) the same parameter

𝜗 is used, but with different range; it turns out that 𝑖_𝑙 is

counterclockwise oriented; instead𝑖_𝑟 is clockwise oriented;

𝑥₀= 𝑖_𝑙,𝑥0(𝜗+

0) = 𝑖𝑟,𝑥0(𝜗

− 0).

Remark 16. The following facts can be derived from the above

equations:

(i) sinceℎ is Lipschitz continuous for every convex body

𝐾, then the involute 𝑖_𝑙,𝑥0is a rectifiable curve;

(ii)𝑖_𝑙,𝑥0(𝜗+₀) = 𝑥₀and

󵄨󵄨󵄨󵄨

󵄨𝑖𝑙,𝑥0(𝜗) − 𝑥𝑙(𝑠𝑙+(𝜗))󵄨󵄨󵄨󵄨󵄨 = 𝑠𝑙+(𝜗) − 𝑠0; (50)

(iii) if𝑥 is a vertex of 𝜕𝐾 then 𝑖_𝑙,𝑥0(𝜗), for (cos 𝜗, sin 𝜗) ∈

𝑁_𝐾(𝑥), lies on an arc of circle centered at 𝑥 with radius

𝑠_𝑙+(𝜗) − 𝑠₀;

(iv) the involute (42) satisfies

𝑖𝑙,𝑥0(𝜗 + 2𝜋) = 𝑖𝑙,𝑥0(𝜗) − 𝐿t𝜗, ∀𝜗 ≥ 𝜗

+

0. (51)

Lemma 17. Parameterization (48) of the involute 𝑖𝑙,𝑥0is 1-1 in

the interval[𝜗₀+, 𝜗+₀ + 2𝜋); moreover, except for at most a finite or countable setF of values 𝜗_𝑖,𝑖 = 1, 2, . . . (corresponding to the 1-face𝐹_𝜃𝑖of𝜕𝐾), 𝑖_𝑙,𝑥0is differentiable and

𝑑

𝑑𝜗𝑖𝑙,𝑥0(𝜗) = (𝑠𝑙+(𝜗) − 𝑠0) n𝜗 𝑓𝑜𝑟 𝜗 > 𝜗

+

0, 𝜗 ∉ F; (52)

furthermore 𝑖_𝑙,𝑥0 has left and right derivative with common directionn_𝜗at𝜗 = 𝜗_𝑖∈ F.

Proof. By differentiating (48) and using (34), equality (52) is

proved. Similar argument, at𝜗 = 𝜗_𝑖∈ F, proves that n_𝜗is the

common direction of the left and right derivatives.

Remark 18. Let𝜗 → 𝑖_𝑙,𝑥1(𝜗), 𝜗 → 𝑖_𝑙,𝑥2(𝜗), 𝑥_𝑖 = 𝑥(𝑠_𝑖), 𝑖 = 1, 2,

be left involutes of𝐾. Since

𝑖_𝑙,𝑥2(𝜗) − 𝑖𝑙,𝑥1(𝜗) = (𝑠2− 𝑠1) t𝜗,

for 𝜗 > max {𝜗+_𝑙 (𝑠₂) , 𝜗+_𝑙 (𝑠₁)} ,

(53)

then they will be called parallel curves. Moreover, by (51),

𝑖_𝑙,𝑥0(𝜗) and 𝑖_𝑙,𝑥0(𝜗 + 2𝜋) will also be called parallel.

Theorem 19. If 𝑑󰜚 is the arc element of the involute 𝑖_𝑙,𝑥0then

𝜗 → 󰜚(𝜗) is continuous and invertible in 𝜗 ≥ 𝜗+₀ with

continuous inverse[0, +∞) ∋ 󰜚 → 𝜗(󰜚). Moreover

𝑑󰜚 = (𝑠𝑙+(𝜗) − 𝑠0) 𝑑𝜗 𝑓𝑜𝑟 𝜗 ≥ 𝜗+0, 𝜗 ∉ F; (54)

the involute is a convex curve with positive curvature a.e.:

𝑑𝜗 𝑑󰜚 = 1 (𝑠𝑙+(𝜗) − 𝑠0) 𝑓𝑜𝑟 𝜗 > 𝜗 + 0, 𝜗 ∉ F, (55) 󰜚 → 𝑖_𝑙,𝑥0(𝜗(󰜚)) is 𝐶1_{everywhere, and} 𝑑 𝑑󰜚𝑖𝑙,𝑥0= n𝜗(󰜚). (56)

Moreover the following properties hold.

(i) For every󰜚 > 0 the right derivative

(𝑑𝜗_𝑑󰜚)+= 1

𝑠_𝑙+(𝜗 (󰜚)) − 𝑠₀ (57)

exists everywhere and it is a decreasing cadlag function.

(ii)(𝑑/𝑑󰜚)𝑖_𝑙,𝑥0has everywhere right derivative given by

(𝑑2

𝑑󰜚2𝑖𝑙,𝑥0)

+

= − 1

𝑠_𝑙+(𝜗 (󰜚)) − 𝑠₀t𝜗(󰜚). (58)

Theorem 20. Let 𝐾(𝑛) _{be a sequence of plane convex bodies}

which converges uniformly to𝐾, 𝑥(𝑛) ∈ 𝜕𝐾(𝑛),𝑥(𝑛) → 𝑥₀; then the corresponding sequences of left involutes𝑖𝐾_𝑙,𝑥(𝑛)(𝑛)converge

uniformly to𝑖_𝑙,𝑥0in compact subsets of[𝜗₀+, +∞]; moreover the corresponding sequence of their derivatives (with respect to the arc length) converges uniformly to(𝑑/𝑑󰜚)𝑖_𝑙,𝑥0.

Proof. By Remark 9 the sequence of functions𝑠𝑛_𝑙+converges

to𝑠_𝑙+. From (54) the arclengths of the left involutes𝑖𝐾_𝑙,𝑥(𝑛)(𝑛)

󰜚(𝑛)(𝜗) = ∫𝜗

𝜗0

(𝑠(𝑛)_𝑙+ (𝜗) − 𝑠(𝑛)₀ ) 𝑑𝜗 (59)

converges uniformly in compact subsets of[𝜗₀+, +∞) to the

arc length󰜚(𝜗) of 𝑖_𝑙,𝑥0; from (56) the same fact holds for their

derivatives.

Let us consider the arc of the involute

𝜂 fl {𝑖_𝑙,𝑥0(𝜗) : 𝜗₀+≤ 𝜗 ≤ 𝜗+₀ + 3𝜋/2} (60)

and the set valued map𝐹 (Definition 8). Let

𝑄 = ⋃ 𝜗+ 0≤𝜗≤𝜗+0+3𝜋/2 {𝜆𝐹 (𝜃) + (1 − 𝜆) 𝑖𝑙,𝑥0(𝜗) , 0 ≤ 𝜆 ≤ 1} , 𝜃 = (cos 𝜗, sin 𝜗) , (61)

the union of segments joining the points of 𝜂 with the

corresponding points on𝜕𝐾.

Definition 21. If the tangent sector𝑇(𝑥₀) to 𝐾 has an opening

less than or equal to𝜋/2 as in Figure 1, then 𝑄 ∪ 𝐾 is convex;

let one define

𝜗∗_𝑙 = 𝜗+₀ + 3𝜋/2. (62)

If𝑄 ∪ 𝐾 is not convex then let us consider co(𝑄 ∪ 𝐾). Let us

notice that𝜕 co(𝑄 ∪ 𝐾) \ 𝜕(𝑄 ∪ 𝐾) is an open segment with

end points𝐴 , 𝐵, with 𝐴 ∈ 𝜂, 𝐵 ∈ 𝜕𝐾. Let us define 𝜗_𝑙∗, with

𝜗₀+ + 3𝜋/2 ≤ 𝜗_𝑙∗ < 𝜗+₀ + 2𝜋 such that (see Figure 2) 𝜃∗_𝑙 =

(cos 𝜗_𝑙∗, sin 𝜗_𝑙∗) is orthogonal to 𝐴𝐵, 𝐵 ∈ 𝐹(𝜃_𝑙∗). Let 𝜗_1,𝑙 be

the smallest𝜃 > 𝜃₀+ satisfying𝐴 = 𝑖_𝑙,𝑥0(𝜗_1,𝑙). Clearly 𝜗∗_𝑙 =

K x0 𝜃+ 0 𝜃− 0 il,x0(𝜗∗l)

Figure 1: Left involute of a square.

For the right involutes a value𝜗_𝑟∗is defined similarly, with

𝜗−

0 − 2𝜋 < 𝜗𝑟∗ ≤ 𝜗0−− 3𝜋/2, such that the line orthogonal to

𝜃∗

𝑟 supporting𝐾 at 𝐹(𝜃∗𝑟) is tangent to the right involute at

𝑖_𝑟,𝑥0(𝜗_1,𝑟) (see Figure 3) where 𝐹(𝜃∗

𝑟) is the point 𝑥(𝑠𝑟−(𝜗∗𝑟)),

written as𝑥(𝜗∗_𝑟) for short.

Theorem 22. Let 𝑖_𝑙:= 𝑖_𝑙,𝑥0be the left involute starting at𝑥₀on the boundary of a plane convex body𝐾; then

(i) the left involute𝜗 → 𝑖_𝑙(𝜗) has the distancing from 𝐾

property for𝜗 ≥ 𝜗₀+but is not SDC for𝜗 ≥ 𝜗∗_𝑙;

(ii) the curve𝜗 ∈ [𝜗+₀, 𝜗∗_𝑙] → 𝑖(𝜗) is SDC;

(iii) for𝑦 ∈ Int(𝐾) the distance function 𝐽_𝑦(𝜗) = |𝑖_𝑙(𝜗) − 𝑦|

is strictly increasing for𝜗 ≥ 𝜗+₀;

(iv) if𝑦 ∈ 𝜕𝐾, then 𝐽_𝑦(𝜗) is not decreasing for 𝜗 ≥ 𝜗₀+and

(𝑑/𝑑𝜗)𝐽 > 0 for (cos 𝜗, sin 𝜗) ∉ 𝑁_𝐾(𝑦).

Proof. As𝑖_𝑙is rectifiable, then the function𝐽_𝑦2(𝜗) = |𝑖_𝑙(𝜗)−𝑦|2

is an absolutely continuous function for𝜗 ≥ 𝜗+₀, and from (52)

for𝜗 ∉ F 1 2 𝑑 𝑑𝜗𝐽𝑦2= ⟨_𝑑𝜗𝑑 𝑖𝑙, 𝑖𝑙(𝜗) − 𝑦⟩ = ⟨(𝑠_𝑙(𝜗) − 𝑠₀) n_𝜗, 𝑥_𝑙(𝑠_𝑙+(𝜗)) + (𝑠_𝑙+(𝜗) − 𝑠₀) t_𝜗− 𝑦⟩ = (𝑠_𝑙+(𝜗) − 𝑠₀) ⟨n_𝜗, 𝑥_𝑙(𝑠_𝑙+(𝜗)) − 𝑦⟩ ≥ 0; (63)

the last inequality holds sincen_𝜗is the outer normal to𝜕𝐾 at

𝑥_𝑙(𝑠_𝑙+(𝜗)). Moreover the previous inequality is strict for all 𝜗

if𝑦 ∈ Int(𝐾), and it is also a strict inequality for 𝑦 ∈ 𝜕𝐾 and

𝑦 ∉ 𝐹(𝜃). This proves (iii) and (iv). Then (i) follows from (iii) and Definition 1 of distancing from K property for a curve. To prove (ii) let us recall that SDC satisfies (2); then one has

to prove that the angle at𝑖_𝑙(𝜗) between the vector 𝑖_𝑙(𝜗) − 𝑖_𝑙(𝜏),

𝜗+

0 < 𝜏 < 𝜗 ≤ 𝜗𝑙∗, andn𝜗, the tangent vector at𝑖𝑙(𝜗), is greater

than or equal to𝜋/2; this is equivalent to show that the half

line𝑟_𝜗through𝑖_𝑙(𝜗) and 𝑥(𝑠_𝑙+(𝜗)) orthogonal to n_𝜗supports

at𝑖_𝑙(𝜗) the arc of 𝑖_𝑙from𝑥₀to𝑖_𝑙(𝜗). By Definition 21 this is the

case for all𝜗 between 𝜗+₀ and𝜗_𝑙∗.

il,x0(𝜗∗l)

K B

x0

A = il(𝜗1,l)

Figure 2: Left involute of an hexagon.

il(𝜗∗r+ 2𝜋) ir(𝜗1,r) x(𝜗∗ r) _{x0 x(𝜗1,r)} K P+(𝜗∗r+ 2𝜋) x(𝜓(𝜗∗ r+ 2𝜋)) y = il,x0( ) = i𝜗l̃ r,x0( )̃𝜗r

Figure 3: Involutes of a circumference.

Corollary 23. The left involute 𝜗 → 𝑖𝑙,𝑥0(𝜗) of the boundary

of a plane convex body𝐾 is a self-distancing curve from 𝐾 for

𝜗 ∈ [𝜗+₀, 𝜗∗_𝑙]; similarly right involute (49) is a self-contracting

curve from𝐾 for 𝜗 ∈ [𝜗_𝑟∗, 𝜗₀−].

Proof. From (i) of Theorem 22 the left involute is a curve such

that the distance of its points from all𝑦 ∈ 𝐾 is not decreasing;

(ii) of the same theorem proves that it is a SDC. Let us recall that a self-contracting curve is a self-distancing curve with opposite orientation.

Theorem 24. Let 𝐾 be a plane convex body not reduced to a

single point and let𝑥₀, 𝑠₀, 𝜃₀, 𝜗₀be the initial parameters. Let

[𝜗+

0, 𝜗0++ 2𝜋] ∋ 𝜗 → 𝑖𝑙(𝜗) be an arc of the left involute starting

at𝑥₀, and let[𝜗−₀ − 2𝜋, 𝜗−₀] ∋ 𝜗 → 𝑖_𝑟(𝜗) be an arc of the right involute ending at𝑥₀; then there exists only one point𝑦 ̸= 𝑥₀ which belongs to both arcs and

𝑦 = 𝑖_𝑙(̃𝜗_𝑙) = 𝑖_𝑟(̃𝜗_𝑟) , (64)

with

𝜗₀−≤ 𝜗_𝑟∗+ 2𝜋 < ̃𝜗𝑙< 𝜗+0+3𝜋₂ ≤ 𝜗∗𝑙,

𝜗_𝑟∗≤ 𝜗₀−−3𝜋₂ < ̃𝜗_𝑟 < 𝜗∗_𝑙 − 2𝜋 ≤ 𝜗₀+.

(65)

Proof. For simplicity, first let us prove the existence of 𝑦

R ∋ 𝜗 → 𝑥(𝜗) := 𝑥(𝜃) defined by (30) is a parameterization

of𝜕𝐾.

Let𝜗 ∈ [𝜗₀, 𝜗₀+ 2𝜋] and let 𝑃+(𝜗) be the first common

point of the half line{𝑥(𝜗) + 𝜆t_𝜗, 𝜆 > 0} and of 𝑖_𝑙. Moreover,

let[𝜗₀, 𝜗∗_𝑙] ∋ 𝜗 → 𝜓(𝜗) be the function satisfying

𝑃+_{(𝜗) = 𝑖}

𝑙(𝜓 (𝜗)) . (66)

Let

𝜙 (𝜗) fl 󵄨󵄨󵄨󵄨𝑃+(𝜗) − 𝑖_{𝑙(𝜗)󵄨󵄨󵄨󵄨 .} (67)

First the following sentence will be proved.

Claim 1.𝑃+(𝜗_𝑙) belongs to 𝑖_𝑟iff the equality

𝜙 (𝜗_𝑙) = 𝐿 (68)

holds for some𝜗_𝑙∈ [𝜗₀, 𝜗₀+ 2𝜋], 𝐿 = |𝜕𝐾|.

Proof of Claim 1. If (68) holds, then

󵄨󵄨󵄨󵄨 󵄨𝑃+(𝜗𝑙) − 𝑥 (𝜗𝑙)󵄨󵄨󵄨󵄨󵄨 =_󵄨𝑃󵄨󵄨󵄨󵄨 +(𝜗𝑙) − 𝑖𝑙(𝜗𝑙)󵄨󵄨󵄨󵄨󵄨 − 󵄨󵄨󵄨󵄨󵄨𝑖𝑙(𝜗𝑙) − 𝑥 (𝜗𝑙)󵄨󵄨󵄨󵄨󵄨 = 𝐿 − 󵄨󵄨󵄨󵄨󵄨arc+(𝑥0, 𝑥 (𝜗𝑙))󵄨󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨arc+(𝑥 (𝜗_𝑙) , 𝑥_{0)󵄨󵄨󵄨󵄨󵄨} = 󵄨󵄨󵄨󵄨󵄨arc−(𝑥₀, 𝑥 (𝜗_{𝑙− 2𝜋))󵄨󵄨󵄨󵄨󵄨.} (69) Thus 𝑃+(𝜗_𝑙) = 𝑥 (𝜗_{𝑙) + 󵄨󵄨󵄨󵄨󵄨arc}+(𝑥 (𝜗_𝑙) , 𝑥_{0)󵄨󵄨󵄨󵄨󵄨t𝜗} 𝑙 = 𝑥 (𝜗𝑙− 2𝜋) + 󵄨󵄨󵄨󵄨󵄨arc−(𝑥0, 𝑥 (𝜗𝑙− 2𝜋))󵄨󵄨󵄨󵄨󵄨t𝜗𝑙−2𝜋 = 𝑖_𝑟(𝜗_𝑙− 2𝜋) . (70)

Thus𝑃+(𝜗_𝑙) is on both arcs of involutes and the other way

around.

Our aim is to prove that there exists𝜗_𝑙 ∈ [𝜗₀, 𝜗₀+ 3𝜋/2]

such that (68) holds. For this goal we prove next Claims 2 and 3.

Claim 2. The following facts hold in[𝜗₀, 𝜗∗_𝑙]:

(i)𝜓 is continuously differentiable and 𝜓󸀠> 0;

(ii)𝜙󸀠> 0.

Proof of Claim 2. Let us prove thatn_𝜗andn_𝜓(𝜗)satisfy

⟨n𝜗, n𝜓(𝜗)⟩ < 0. (71)

Let us consider the triangle with vertices 𝑥(𝜗), 𝑖_𝑙(𝜓(𝜗)),

𝑥(𝜓(𝜗)). As

󵄨󵄨󵄨󵄨𝑖𝑙(𝜓 (𝜗)) − 𝑥 (𝜓 (𝜗))󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨arc+(𝑥0, 𝑥 (𝜓 (𝜗)))󵄨󵄨󵄨󵄨

≥ 󵄨󵄨󵄨󵄨arc+(𝑥 (𝜗) , 𝑥 (𝜓 (𝜗)))󵄨󵄨󵄨󵄨 ≥󵄨󵄨󵄨󵄨𝑥(𝜓(𝜗)) − 𝑥(𝜗)󵄨󵄨󵄨󵄨, (72)

the angle between𝑥(𝜓(𝜗)) − 𝑃+(𝜗) and 𝑥(𝜗) − 𝑃+(𝜗) is acute

and the angle between n_𝜗 and n_𝜓(𝜗) is obtuse. Thus (71)

follows. By definition, 𝜓(𝜗) solves (66); thus 𝜓(𝜗) is the

implicit solution to

⟨𝑖_𝑙(𝜓 (𝜗)) − 𝑥 (𝜗) , n𝜗⟩ = 0. (73)

As

⟨_𝑑𝜓𝑑 𝑖_𝑙(𝜓) , n_𝜗⟩ = (𝑠 (𝜓) − 𝑠₀) ⟨n_𝜓, n_𝜗⟩ (74)

is negative by (71), then by Dini’s Theorem equation (73) has

a solution𝜓(𝜃) satisfying

(𝑠 (𝜓) − 𝑠₀) ⟨n_𝜓, n_𝜗⟩ 𝜓󸀠(𝜗) + ⟨𝑖_𝑙(𝜓 (𝜗)) − 𝑥 (𝜗) , t𝜗⟩

= 0. (75)

As𝑖_𝑙(𝜓(𝜗)) − 𝑥(𝜗) = 𝜆t_𝜗(𝜆 > 0) and (71) holds, then 𝜓󸀠> 0,

and𝜓 is strictly increasing and continuously differentiable.

Let us prove (ii). The formula 𝑑 𝑑𝜗󵄨󵄨󵄨󵄨𝑖𝑙(𝜗) − 𝑖𝑙(𝜓 (𝜗))󵄨󵄨󵄨󵄨 2 = 2 ⟨𝑖_𝑙(𝜗) − 𝑖_𝑙(𝜓 (𝜗)) ,_𝑑𝜗𝑑 𝑖_𝑙(𝜗) − 𝑑 𝑑𝜗𝑖𝑙(𝜓 (𝜗))⟩ (76)

holds. Let us notice that𝑖_𝑙(𝜗) − 𝑖_𝑙(𝜓(𝜗)) is parallel to t_𝜗; thus

by (52)

⟨𝑖_𝑙(𝜗) − 𝑖𝑙(𝜓 (𝜗)) ,_𝑑𝜗𝑑 𝑖𝑙(𝜗)⟩ = 0. (77)

On the other hand

− ⟨𝑖_𝑙(𝜗) − 𝑖𝑙(𝜓 (𝜗)) ,_𝑑𝜗𝑑 𝑖𝑙(𝜓 (𝜗))⟩

= − ⟨−𝑠 (𝜗) t𝜗− 𝜆t𝜗, (𝑠 (𝜓 (𝜗)) − 𝑠0) n𝜓(𝜗)⟩ 𝜓󸀠

= (𝑠 (𝜗) + 𝜆) (𝑠 (𝜓 (𝜗)) − 𝑠0) ⟨t𝜗, n𝜓(𝜗)⟩ 𝜓󸀠.

(78)

As the angle betweent_𝜗andn_𝜓(𝜗)is acute, then last term in

the above equalities is positive; thus the derivative in the left hand side of (76) is positive and (ii) of Claim 2 follows.

Claim 3. In the interval[𝜗₀, 𝜗∗_𝑙] the function 𝜙 has values

smaller than𝐿 and greater than 𝐿.

Proof of Claim 3. The angles𝜗∗_𝑟 and𝜗_1,𝑟have been introduced

in Definition 21. For simplicity𝑥(𝑠_𝑟−(𝜗_1,𝑟)) will be denoted

with 𝑥(𝜗_1,𝑟). Let us consider the convex set bounded by

arc+(𝑥(𝜓(𝜗_𝑟∗+ 2𝜋)), 𝑥(𝜗_1,𝑟)) and by the polygonal line with

vertices 𝑥(𝜗_1,𝑟), 𝑖_𝑟(𝜗_1,𝑟), 𝑃+(𝜗∗_𝑟 + 2𝜋), 𝑥(𝜓(𝜗∗_𝑟 + 2𝜋)); see

Clearly the inequalities 󵄨󵄨󵄨󵄨𝑖𝑟(𝜗1,𝑟) − 𝑃+(𝜗𝑟∗+ 2𝜋)󵄨󵄨󵄨󵄨 < 󵄨󵄨󵄨󵄨𝑖𝑟(𝜗1𝑟) − 𝑥 (𝜗1,𝑟)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨arc−_{(𝑥 (𝜗} 1,𝑟) , 𝑥 (𝜓 (𝜗∗𝑟 + 2𝜋)))󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑥 (𝜓 (𝜗∗ 𝑟 + 2𝜋)) − 𝑃+(𝜗𝑟∗+ 2𝜋)󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨arc−_(𝑥 0, 𝑥 (𝜗1,𝑟))󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨arc−_{(𝑥 (𝜗} 1,𝑟) , 𝑥 (𝜓 (𝜗∗𝑟 + 2𝜋)))󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨arc−_{(𝑥 (𝜓 (𝜗}∗ 𝑟 + 2𝜋)) , 𝑥0)󵄨󵄨󵄨󵄨 = 𝐿 (79) hold. As 𝜙 (𝜗_𝑟∗_{+ 2𝜋) = 󵄨󵄨󵄨󵄨𝑖𝑙}(𝜗∗_𝑟 + 2𝜋) − 𝑃+(𝜗_𝑟∗_{+ 2𝜋)󵄨󵄨󵄨󵄨} < 󵄨󵄨󵄨󵄨𝑖𝑟(𝜗1,𝑟) − 𝑃+(𝜗∗𝑟 + 2𝜋)󵄨󵄨󵄨󵄨 , (80) using the previous inequalities, one obtains

𝜙 (𝜗_𝑟∗+ 2𝜋) < 𝐿. (81)

Let us show now that

𝜙 (𝜗₀+3𝜋

2 ) > 𝐿 (82)

holds.

Let𝜌 be the half line with origin 𝑥₀and direction−t_𝜗0;

𝜌 − {𝑥₀} crosses the arc 𝑖_𝑟 in a first point𝑦₁ = 𝑖_𝑟(𝛼₁), with

𝛼₁< 𝜗₀− 𝜋/2. Then

𝑟 fl 󵄨󵄨󵄨󵄨𝑥0− 𝑦1󵄨󵄨󵄨󵄨 < 󵄨󵄨󵄨󵄨𝑦1− 𝑥 (𝛼1)󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨arc−(𝑥 (𝛼1) , 𝑥0)󵄨󵄨󵄨󵄨

= 𝐿. (83)

The half line𝜌 meets the arc 𝑖_𝑙in a point𝑦₂and|𝑦₂− 𝑥₀| = 𝐿.

Property (iii) of Theorem 22 implies that the arc 𝐷 of

the left involute after𝑦₂lies outside of the circle centered in

𝑥₀and with radius𝐿. Similar property for the right involute

implies that the arc𝐶 of the right involute joining 𝑥₀to𝑦₁lies

in the circle with center𝑥₀and radius𝑟; thus the straight line

tangent to𝐾 at 𝑥(𝜗₀+ 3𝜋/2) meets the arc 𝐶 in 𝑖_𝑟(𝜗₀− 𝜋/2)

and𝐷 in 𝑃+(𝜗₀+ 3𝜋/2). Therefore 𝜙 (𝜗0+_2𝜋3 ) =󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑖𝑙(𝜗0+3𝜋₂ ) − 𝑃+(𝜗0+3𝜋₂ )󵄨󵄨󵄨󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨𝑖𝑙}(𝜗₀+3𝜋₂ ) − 𝑥 (𝜗₀+3𝜋₂ )󵄨󵄨󵄨󵄨_󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨𝑥(𝜗0}+3𝜋 2 ) − 𝑃+(𝜗0+ 3𝜋 2 )󵄨󵄨󵄨󵄨󵄨󵄨󵄨 >󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨𝑖𝑙}(𝜗₀+3𝜋 2 ) − 𝑥 (𝜗0+ 3𝜋 2 )󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨𝑥(𝜗0}+3𝜋 2 ) − 𝑖𝑟(𝜗0− 𝜋 2)󵄨󵄨󵄨󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨arc}+(𝑥0, 𝑥 (𝜗0+3𝜋₂ ))󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨arc}−(𝑥₀, 𝑥 (𝜗₀+3𝜋₂ ))󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨 = 𝐿.} (84) Inequality (82) is proved.

The intermediate values theorem implies that there exists

𝜗𝑙∈ [2𝜋 + 𝜗∗𝑟, 𝜗0+ 3𝜋/2] such that (68) holds. Claim 1 implies

that

𝑃+(𝜗_𝑙) = 𝑖_𝑙(𝜓 (𝜗_𝑙)) = 𝑖_𝑟(𝜗_𝑙− 2𝜋) , (85)

so the right involute and the left involute meet each other in

one point and (64) is proved with ̃𝜗_𝑙= 𝜓(𝜗_𝑙), ̃𝜗_𝑟 = 𝜗_𝑙− 2𝜋.

By approximation argument the same result holds for an

arbitrary convex body𝐾.

Let us prove now that the point𝑦 is unique. Let us argue

by contradiction. Let𝑃, 𝑄 be two distinct points on 𝑖_𝑙∩𝑖_𝑟, with

𝑃 ≺ 𝑄 on 𝑖_𝑙and𝑖_𝑟; then since𝑖_𝑙is a distancing curve from𝑥₀,

󵄨󵄨󵄨󵄨𝑃 − 𝑥0󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑄 − 𝑥0󵄨󵄨󵄨󵄨, (86)

and since𝑖_𝑟is a contracting curve to𝑥₀,

󵄨󵄨󵄨󵄨𝑃 − 𝑥0󵄨󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨󵄨𝑄 − 𝑥0󵄨󵄨󵄨󵄨, (87)

therefore all the points on the arc of 𝑖_𝑙 and of 𝑖_𝑟 between

𝑃 and 𝑄 have the same distance from 𝑥₀; thus, between𝑃

and𝑄, 𝑖_𝑙 and𝑖_𝑟 (arc of involutes of a same convex body𝐾)

coincide with the same arc of circle centered at𝑥₀; this implies

that𝐾 reduce to the point 𝑥₀, which is not possible for the

assumption.

Definition 25. Let𝑧 ∉ 𝐾. Let 𝑧_𝑙(𝑧_𝑟) ∈ 𝜕𝐾 on the contact set on

the “left” (right) support line to𝐾 through 𝑧. If the contact set

is a 1-face on these support lines, then𝑧_𝑙and𝑧_𝑟are identified

as the closest ones to𝑧. The triangle 𝑧𝑧_𝑙𝑧_𝑟is counterclockwise

oriented.

Theorem 26. For every 𝜉 ∈ 𝜕𝐾 let one consider the left

involutes𝑖_𝑙,𝜉and the right involutes𝑖_𝑟,𝜉parameterized by their arc length󰜚. The maps

𝜕𝐾 × (0, +∞) ∋ (𝜉, 󰜚) 󳨀→ 𝑖_𝑙,𝜉(𝜃 (󰜚)) ∈ R2\ 𝐾, 𝜕𝐾 × (0, +∞) ∋ (𝜉, 󰜚) 󳨀→ 𝑖_𝑟,𝜉(𝜃 (󰜚)) ∈ R2\ 𝐾 (88) are 1-1 maps.

Proof. Assume, in the proof, that𝑥₀ ∈ 𝜕𝐾, 𝜃₀ ∈ 𝐺(𝑥₀), 𝜗₀, 𝑠₀

are fixed. Let𝑧 ∉ 𝐾. The tangent sector to the cap body 𝐾𝑧

with vertex z has two maximal segments𝑧𝑧_𝑙and𝑧𝑧_𝑟on the

sides that do not meet𝐾 (except at the end points 𝑧_𝑙and𝑧_𝑟).

Let𝜗_𝑙be such that𝑧_𝑙= 𝑥_𝑙(𝑠_𝑙+(𝜗_𝑙)), and let 𝑠 be such that

󵄨󵄨󵄨󵄨𝑧 − 𝑧𝑙󵄨󵄨󵄨󵄨 = 𝑠𝑙+(𝜗𝑙) − 𝑠. (89)

Let𝜉_𝑙= 𝑥_𝑙(𝑠); let 𝜗 = 𝜗+_𝑙(𝑠). From (50) and from the definition

of left involute (46) (with𝜉_𝑙in place of𝑥₀,𝜗 in place of 𝜗₀+, and

𝑠 in place 𝑠₀)

holds; thus the map(𝜉, 󰜚) → 𝑖_𝑙,𝜉(󰜚) is surjective. Moreover the map is also injective, since the left involutes do not cross each other since they are parallel (see Remark 18). Similar proof holds for the right involutes.

Let𝜉_𝑙 = 𝑥_𝑙(𝑠) be the starting point of the left involute 𝑖_{𝑙,𝜉𝑙}

through𝑧, defined in the previous theorem; similarly let 𝜉_𝑟be

the starting point of the right involute𝑖_{𝑟,𝜉𝑟}through𝑧. Let us

notice that𝑖_{𝑙,𝜉𝑙}and𝑖_{𝑟,𝜉𝑟}meet each other in a countable ordered

set of points.

3.1. J-Fence and G-Fence

Definition 27. Let𝐾 be a convex body in R2,|𝜕𝐾| > 0, 𝑥₀ ∈

𝜕𝐾, 𝜃₀∈ 𝐺(𝑥₀), 𝜃₀= (cos 𝜗₀, sin 𝜗₀), 𝑠₀∈ R. Let 𝑖_𝑙fl 𝑖_𝑙,𝑥0, and

let𝑖_𝑟fl 𝑖_𝑟,𝑥0. Let

𝑦 = 𝑖_𝑙(̃𝜗_𝑙) = 𝑖_𝑟(̃𝜗_𝑟) ∈ R2\ 𝐾 (91)

be the first point where the two involutes cross each other (see Theorem 24). Let one define

J𝑙(𝐾, 𝑥0) fl {𝑦 ∈ R2: 𝑦 = 𝑡𝑥0+ (1 − 𝑡) 𝑖𝑙(𝜗) , 0 ≤ 𝑡 ≤ 1, 𝜗+₀ ≤ 𝜗 ≤ ̃𝜗𝑙} , J𝑟(𝐾, 𝑥0) fl {𝑦 ∈ R2: 𝑦 = 𝑡𝑥0+ (1 − 𝑡) 𝑖𝑟(𝜗) , 0 ≤ 𝑡 ≤ 1, ̃𝜗𝑟≤ 𝜗 ≤ 𝜗−0} , J (𝐾, 𝑥0) fl (J𝑙(𝐾, 𝑥0) ∪ J𝑟(𝐾, 𝑥0)) \ Int (𝐾) . (92)

J(𝐾, 𝑥₀) will be called the J-fence of 𝐾 at 𝑥₀.

Let us notice thatJ_𝑙(𝐾, 𝑥₀) and J_𝑟(𝐾, 𝑥₀) are two convex

bodies in common with the segment𝑥₀𝑦 only.

From Theorem 26 the starting point𝜉_𝑙(𝜉_𝑟) of a left (right)

involute is uniquely determined from any point𝑧 ∉ 𝐾 of

the involute. The arc of the points on the left (right) involute

between the starting point and𝑧 will be denoted by 𝑖𝑧_𝑙,𝜉

𝑙(𝑖

𝑧 𝑟,𝜉𝑟)

or𝑖𝑧_𝑙 (𝑖𝑧_𝑟) for short. For𝑦 ⪯ 𝑤 let us denote with 𝑖𝑦,𝑤_𝑙 (𝑖𝑦,𝑤_𝑟 ) the

oriented arc of the left (right) involute between𝑦 and 𝑤.

Let us introduce now other regions which are bounded by left and right involutes.

Let us fix the initial parameters𝑥₀, 𝑠₀, 𝜃₀, 𝜗₀.

Definition 28. Given𝑧 ∈ R2 \ 𝐾, let 𝑖_𝑙 = 𝑖_{𝑙,𝜉𝑙} (𝑖_𝑟 = 𝑖_{𝑟,𝜉𝑟}) be

the left (right) involute through𝑧 with starting point 𝜉_𝑙(𝜉_𝑟)

and let𝑧_𝑙 (𝑧_𝑟) ∈ 𝜕𝐾 be as in Definition 25. Let 𝜗_𝜉+

𝑙satisfying

𝑥𝑙(𝑠𝑙+(𝜗+𝜉𝑙)) = 𝜉𝑙. Let𝜗𝑙 > 𝜗

+

𝜉𝑙 be the smallest angle for which

𝑥𝑙(𝑠𝑙+(𝜗𝑙)) = 𝑧𝑙. Let one consider the parameterization (46);

let one define

G_𝑙(𝐾, 𝑧) fl {𝑡𝑥𝑙(𝑠𝑙+(𝜗)) + (1 − 𝑡) 𝑖𝑙(𝜗) , 0 < 𝑡

< 1, 𝜗+_𝜉𝑙 < 𝜗 < 𝜗_𝑙} . (93)

If𝑖𝑧_𝑙 does not cross the open segment𝑧𝑧_𝑙, the regionG_𝑙(𝐾, 𝑧)

is an open set bounded by the convex arc of left involute𝑖𝑧_𝑙, the

segment𝑧𝑧_𝑙, and the convex arc of𝜕𝐾: arc+(𝜉_𝑙, 𝑧_𝑙); otherwise

let𝑤 be the nearest point to 𝑧 where 𝑖𝑧_𝑙 crosses the open

segment𝑧𝑧_𝑙; the regionG_𝑙(𝐾, 𝑧) is an open set bounded by

the arc𝑖𝑤,𝑧_𝑙 , the segment𝑤𝑧, and 𝜕𝐾. Similarly let us define

G_𝑟(𝐾, 𝑧).

G_𝑙(𝐾, 𝑧), G_𝑟(𝐾, 𝑧) are open and bounded sets. Let us

define

G (𝐾, 𝑧) fl Int (cl (G𝑙(𝐾, 𝑧) ∪ G𝑟(𝐾, 𝑧))) . (94)

G(𝐾, 𝑧) is an open, bounded, connected set. G(𝐾, 𝑧) will be

called theG-fence of 𝐾 at 𝑧.

Remark 29. If𝑧 is the first crossing point of 𝑖_𝑙and𝑖_𝑟and𝜉_𝑙=

𝜉𝑟, thenG(𝐾, 𝑧) = Int(J(𝐾, 𝜉𝑙)).

Let us conclude this section with the following result, which follows from Theorem 20.

Theorem 30. Let 𝐾 be limit of a sequence of convex bodies

𝐾(𝑛)_,𝑥

0= lim 𝑥(𝑛)0 , and𝑥(𝑛)0 ∈ 𝜕𝐾(𝑛). Then

J (𝐾(𝑛), 𝑥(𝑛)₀ ) 󳨀→ J (𝐾, 𝑥₀) . (95)

Moreover if𝑧 ∉ 𝐾, 𝑧 = lim 𝑧(𝑛),𝑧(𝑛)∉ 𝐾(𝑛), then

cl(G (𝐾(𝑛), 𝑧(𝑛))) 󳨀→ cl (G (𝐾, 𝑧)) . (96)

4. Bounding Regions for SDC in the Plane

Let us assume that𝑥₀is the end point of one of the following

sets:

(a) a steepest descent curve𝛾;

(b)𝛾𝐾: a self-distancing curve from a convex body𝐾; see

Definition 3.

The following questions arise: can one extend𝛾, 𝛾𝐾beyond

𝑥₀? Which regions delimit that extension? Which regions are

allowed and which are forbidden?

Lemma 31. Let 𝑧 ∈ R2_{\ 𝐾. If 𝑢 ∈ G}

𝑙(𝐾, 𝑧) then the arc 𝑖𝑙𝑢

of the left involute to𝐾 ending at 𝑢 is contained in G_𝑙(𝐾, 𝑧). Similarly if𝑢 ∈ G_𝑟(𝐾, 𝑧), then 𝑖𝑢_𝑟 ⊂ G_𝑟(𝐾, 𝑧).

Proof. Since𝑢 ∈ G_𝑙(𝐾, 𝑧), by (93) there exist 𝜗_𝑙 ∈ (𝜗+_𝜉𝑙, 𝜗_𝑙),

𝜏 ∈ (0, 1) such that

𝑢 = 𝜏𝑥𝑙(𝑠𝑙+(𝜗𝑙)) + (1 − 𝜏) 𝑖𝑙(𝜗𝑙) . (97)

Then the arc𝑖_𝑙𝑢 is parallel to an arc of the left involute 𝑖_𝑙

(through𝑧) for 𝜗 ∈ (𝜗+_𝜉

𝑙, 𝜗𝑙). Then any left tangent segment to

𝐾 from a point of 𝑖𝑢

𝑙 is contained in the left tangent segment

Lemma 32. Let 𝑧 ∈ R2_{\ 𝐾 and let 𝑢 ∈ G}

𝑙(𝐾, 𝑧). There are

two possible cases:

(i) if the right involute ending at 𝑢 does not cross the

tangent segment 𝑧_𝑙𝑧 or it crosses 𝑧_𝑙𝑧 at a point 𝑞 ∈

G_𝑙(𝐾, 𝑧), then in both cases 𝑖𝑢

𝑟 ⊂ G𝑙(𝐾, 𝑧);

(ii) if the right involute ending at 𝑢 crosses the tangent

segment 𝑧_𝑙𝑧 at a point 𝑞 ∈ 𝑧_𝑙𝑧 ∩ 𝜕G_𝑙(𝐾, 𝑧), then

𝑖𝑞,𝑢

𝑟 \ {𝑞} ⊂ G𝑙(𝐾, 𝑧).

Proof. Since the starting point 𝜉_𝑟(𝑢) of the right involute

ending at𝑢 is on 𝜕𝐾, the distance from 𝜉_𝑟(𝑢) to a point of

the left involute𝑖𝑧_𝑙 is not decreasing; see (iv) of Theorem 22;

similarly the distance from 𝜉_𝑟(𝑢) to a point of 𝑖𝑢_𝑟 is not

decreasing. In case (i) the arc𝑖𝑢_𝑟has its end points inG_𝑙(𝐾, 𝑧)

and by the above distance property it can not cross two times the left involute; then it can not cross the boundary

ofG_𝑙(𝐾, 𝑧); therefore 𝑖𝑢_𝑟 ⊂ G_𝑙(𝐾, 𝑧); similarly in case (ii) the

arc𝑖𝑞,𝑢_𝑟 can not cross the boundary ofG_𝑙(𝐾, 𝑧) at most in 𝑞;

therefore all the points of this arc, except to that𝑞, belong to

G_𝑙(𝐾, 𝑧).

From the previous lemma the following follows.

Theorem 33. Let 𝑧 ∉ 𝐾. The following inclusions hold:

(a) if𝑢 ∈ G_𝑙(𝐾, 𝑧), then cl(G_𝑙(𝐾, 𝑢)) \ 𝜕𝐾 ⊂ G_𝑙(𝐾, 𝑧) ; (98) (b) if𝑢 ∈ G_𝑟(𝐾, 𝑧), then cl(G_𝑟(𝐾, 𝑢)) \ 𝜕𝐾 ⊂ G_𝑟(𝐾, 𝑧) ; (99) (c) if𝑢 ∈ G(𝐾, 𝑧), then cl(G (𝐾, 𝑢)) \ 𝜕𝐾 ⊂ G (𝐾, 𝑧) . (100)

Proof. By Lemma 31 the left involute that boundsG_𝑙(𝐾, 𝑢) is

insideG_𝑙(𝐾, 𝑧); then (98) is proved. Inclusion (99) is proved

similarly. Let𝑢 ∈ G(𝐾, 𝑧) = Int(cl(G_𝑙(𝐾, 𝑧) ∪ G_𝑟(𝐾, 𝑧))) and

let us consider𝑢 ∈ G_𝑙(𝐾, 𝑧); then in case (i) of Lemma 32

also the open arc of the right involute𝑖𝑢_𝑟 is insideG_𝑙(𝐾, 𝑧) ⊂

G(𝐾, 𝑧). Besides 𝑖𝑢

𝑙 ⊂ 𝜕G𝑙(𝐾, 𝑢); then (100) is trivial. In case

(ii) of Lemma 32 the open arc𝑖𝑞,𝑢_𝑟 is insideG_𝑙(𝐾, 𝑧). On the

other hand𝑞 is inside G_𝑟(𝐾, 𝑧) and by (99) the arc 𝑖𝑞_𝑟 ⊂ 𝑖𝑢_𝑟 is in

G_𝑟(𝐾, 𝑧) ⊂ G(𝐾, 𝑧). Similar arguments hold if 𝑢 ∈ G_𝑟(𝐾, 𝑧).

Then (100) holds in this case too.

Lemma 34. Let 𝑤 ∉ 𝐾. Let 𝜂 be polygonal deleted SDC_𝐾with end point𝑦 ∈ G(𝐾, 𝑤). Then

𝜂 ⊂ G (𝐾, 𝑤) , (101)

𝜂 ⊂ cl (G (𝐾, 𝑦)) . (102)

Proof. To prove (101), let us assume, by contradiction, that𝜂

has a point𝑧 ∉ G(𝐾, 𝑤). With no loss of generality it can be

assumed that𝑧 ∈ 𝜕G(𝐾, 𝑤) and

𝜂 \ 𝜂_𝑧⊂ G (𝐾, 𝑤) . (103)

Then, 𝑧 is the end point of a segment 𝑧𝑤_𝑖, where 𝑤_𝑖 ∈

G(𝐾, 𝑤) ∩ 𝜂 and 𝑧 ≺ 𝑤_𝑖on𝜂. As 𝑧 ∈ 𝜕G(𝐾, 𝑤), then there

exists an involute through𝑧 which is a piece of the boundary

ofG(𝐾, 𝑤) (to fix the ideas it is assumed that it is the left

involute𝑖_𝑙). Let us consider𝑧_𝑙∈ 𝜕𝐾 so that the tangent vector

t_𝑧to𝑖_𝑙at𝑧 satisfies

⟨t_𝑧, 𝑧 − 𝑧_𝑙⟩ = 0. (104)

As𝑤_𝑖is inside the orthogonal angle centered in𝑧 with sides

t_𝑧and𝑧_𝑙− 𝑧, then

⟨𝑤𝑖− 𝑧, 𝑧 − 𝑧𝑙⟩ < 0. (105)

Then as for𝜀 > 0 sufficiently small, 𝑧_𝜀 := 𝑧 + 𝜀(𝑤_𝑖− 𝑧) ∈ 𝜂_𝑤𝑖

and at𝑧_𝜀the curve𝜂 has tangent vector 𝑤_𝑖− 𝑧 that satisfies

⟨𝑤_𝑖− 𝑧, 𝑧_𝜀− 𝑧_𝑙⟩ < 0, (106)

contradicting the fact that 𝜂_𝑤 has the distancing from 𝐾

property (5). This proves (101).

If𝑤_𝑛 → 𝑦, with 𝑦 ∈ G(𝐾, 𝑤_𝑛), also the inclusions

𝜂 ⊂ cl (G (𝐾, 𝑤_𝑛)) (107)

hold. Then (102) is obtained by approximation Theorem 30.

Theorem 35. Let 𝐾 be a convex body and let 𝛾𝐾be SDC_𝐾,

𝑤 ∈ 𝛾, 𝑤 ∉ 𝐾. Then

𝛾_𝑤𝐾⊂ cl (G (𝐾, 𝑤)) . (108)

Proof. Let us choose a sequence{𝑤_𝑛}, 𝑤_𝑛∈ 𝛾𝐾, 𝑤_𝑛⪯ 𝑤, 𝑤_𝑛→

𝑤. Let us fix the arc 𝛾𝐾

𝑤𝑛. By [8, Theorem 6.16],𝛾

𝐾

𝑤𝑛is limit of

polygonal SDC_𝐾with end point𝑤_𝑛. From Lemma 34, these

polygonal SDC_𝐾are enclosed in cl(G(𝐾, 𝑤_𝑛)); then

𝛾_𝑤𝐾_𝑛 ⊂ cl (G (𝐾, 𝑤𝑛)) (109)

holds too. Inclusion (108) is now obtained from the limit of the previous inclusions and by the approximation Theorem 30.

Theorem 36. Let 𝐾 be a convex body not reduced to a point.

If𝛾𝐾is a self-distancing curve from𝐾 with starting point 𝑥₀∈

𝜕𝐾, then

𝛾𝐾⊂ cl (R2\ (J (𝐾, 𝑥0) ∪ 𝐾)) . (110)

Proof. Let𝑧 be the first crossing point of the left and right

involutes of𝐾 starting at 𝑥₀. Then

Int(J (𝐾, 𝑥₀)) = G (𝐾, 𝑧) . (111)

By contradiction, if𝛾𝐾 has a point𝑤 ∈ G(𝐾, 𝑧), then, by

Theorem 35, the following inclusion holds:

since, by the distancing from𝐾 property, 𝛾𝐾has in common

with𝐾 only the starting point 𝑥₀then the inclusion

𝛾𝐾_𝑤\ {𝑥₀} ⊂ cl (G (𝐾, 𝑤)) \ 𝜕𝐾 (113)

holds too. Moreover by (100) the set cl(G(𝐾, 𝑤)) \ 𝜕𝐾 has

positive distance from theR2\ G(𝐾, 𝑧); then 𝛾_𝑤𝐾\ {𝑥₀} has

a positive distance fromR2\ G(𝐾, 𝑧) = R2\ Int(J(𝐾, 𝑥₀)).

This is in contradiction with𝑥₀∈ 𝜕J(𝐾, 𝑥₀).

Corollary 37. Let 𝛾 be SDC and let 𝑧₁∈ 𝛾; then

𝛾 \ 𝛾_𝑧1⊂ cl (R2_{\ J (co (𝛾}

𝑧1) , 𝑧1)) . (114)

Proof. Since𝛾 \ 𝛾_𝑧1is a self-distancing curve from co(𝛾_𝑧1) and

𝑧₁ ∈ 𝜕 co(𝛾_𝑧1) (see [8, (i) of Lemma 4.6]), then Theorem 36

applies to𝛾𝐾= 𝛾 \ 𝛾_𝑧1with𝐾 = co(𝛾_𝑧1).

Definition 38. Let𝛾 be SCD. If 𝑧₁, 𝑧 ∈ 𝛾, with 𝑧₁⪯ 𝑧 let

𝛾𝑧1,𝑧fl 𝛾𝑧\ 𝛾𝑧1. (115)

For𝑧 ∉ 𝐾, let 𝐾𝑧be the cap body, introduced in (9). Next

theorem shows the principal result on bounding regions for

arcs of SDC𝛾.

Theorem 39. Let 𝐾 be a convex body and let 𝛾 be SDC_𝐾. If

𝑧₁, 𝑧 ∈ 𝛾, with 𝑧₁⪯ 𝑧 then

𝛾_𝑧1,𝑧⊂ cl (G (𝐾, 𝑧) \ J (𝐾𝑧1_{, 𝑧}

1)) . (116)

Proof. First let us notice that𝛾_𝑧1,𝑧has the distancing from𝐾

and from the set point{𝑧₁} property; thus by Proposition 7

it has the distancing from𝐾𝑧1_{property. Then inclusion (116)}

follows from Theorems 35 and 36.

Let us conclude the section with the following inclusion

result forJ-fences.

Theorem 40. Let 𝐾, 𝐻 be two convex bodies not reduced to a

point,𝐾 ⊂ 𝐻. Let 𝑥₀∈ 𝜕𝐾 ∩ 𝜕𝐻. Then

J (𝐾, 𝑥₀) ⊂ J (𝐻, 𝑥₀) . (117)

Proof. The boundary of J(𝐻, 𝑥₀) consists of two arcs

of the left and right involutes of 𝐻 starting at 𝑥₀. By

Corollary 23 they are SDC_𝐻, and then they are SDC_𝐾;

therefore by Theorem 36 they cannot intersect the boundary

ofJ(𝐾, 𝑥₀).

4.1. Minimally Connecting Plane Steepest Descent Curves.

Given a point 𝑥₁ ∉ 𝐾, the segment joining it with its

projection𝑥₀on𝜕𝐾 is SDC_𝐾which minimally connects the

two points.

This subsection is devoted to consider when it would be

possible to connect a given point𝑥₀ on the boundary of a

plane convex body𝐾, with an arbitrarily given point 𝑥₁∉ 𝐾,

by using a steepest descent curve𝛾 ∈ SDC_𝐾. Let us denote

withΓ_𝑥𝐾_0,𝑥1the class of the curves𝛾 ∈ SDC_𝐾starting at𝑥₀and

ending at𝑥₁. N Kx0 Pl Br V P yr y∗ r yl x1 ̃iP𝑟 r,x0 ̃iP𝑙 l,x0 _Ql

Figure 4: The regions𝑁, 𝐵_𝑟, and𝑉 when 𝐾 is a square.

Definition 41. Let𝛾 be SDC with end point 𝑦 and let 𝜂 be SDC

with starting point𝑦; let us denote by 𝛾 ∗ 𝜂 the curve joining

𝛾 with 𝜂 in the natural order, if it is SDC curve.

Theorem 42. Let 𝑥₀∈ 𝜕𝐾, 𝑥₁∉ 𝐾. Then Γ_𝑥𝐾_0,𝑥1 ̸= 0 iff

𝑥₁∈ cl (R2\ (J (𝐾, 𝑥₀)) ∪ 𝐾) . (118)

If (118) holds, there exist at most two𝜂_𝑖 ∈ Γ_𝑥𝐾

0,𝑥1,𝑖 = 1, 2 such

that the following properties are true:

∀𝛾 ∈ Γ_𝑥𝐾_0,𝑥1 󳨐⇒ co(𝜂₁) ⊂ co (𝛾) 𝑜𝑟 co (𝜂₂) ⊂ co (𝛾) , (𝑜𝑟 𝑏𝑜𝑡ℎ), (119) ∀𝛾 ∈ Γ_𝑥𝐾_0,𝑥1 󳨐⇒ 󵄨󵄨󵄨󵄨𝛾󵄨󵄨󵄨󵄨 ≥ min_𝑖=1,2{󵄨󵄨󵄨󵄨𝜂𝑖󵄨󵄨󵄨󵄨}. (120) Proof. Let𝛾 ∈ Γ_𝑥𝐾

0,𝑥1. From (110) of Theorem 36, since𝑥1∈ 𝛾,

then (118) follows.

Let us prove now that (118) is sufficient. Let us notice that

R2\ (J(𝐾, 𝑥₀) ∪ 𝐾) can be divided into four regions 𝑁, 𝐵_𝑙,

𝐵_𝑟, and𝑉 (see Figure 4) defined as follows:

(i) the closed normal sector𝑁 := 𝑥₀+𝑁_𝐾(𝑥₀) is the angle

bounded by the two half lines𝑡_𝑙, 𝑡_𝑟 tangent at𝑥₀ ∈

𝜕𝐾 to the left and right involute 𝑖𝑙 := 𝑖𝑙,𝑥0,𝑖𝑟 := 𝑖𝑟,𝑥0,

respectively; this angle can be reduced to an half line,

starting at𝑥₀;

(ii) let𝑃 be the first crossing point between 𝑖_𝑙and𝑖_𝑟; see

Theorem 24;𝑖_𝑙is SDC to𝑖_𝑙(𝜗∗_𝑙), which will be a point

𝑄_𝑙following𝑃; after 𝑄_𝑙the involute𝑖_𝑙is no more SDC;

see (i) of Theorem 22.

Let us change𝑖_𝑙after𝑄_𝑙with𝑗_{𝑙,𝑄𝑙}, the left involute of

co(𝐾 ∪ 𝑖𝑄𝑙

𝑙 ) at 𝑄𝑙.

Let us define𝑃_𝑙as the first intersection point of𝑗_{𝑙,𝑄𝑙}

with𝜕𝑁, and let

̃𝑖𝑃𝑙 𝑙,𝑥0fl 𝑖 𝑄𝑙 𝑙,𝑥0∗ 𝑗 𝑃𝑙 𝑙,𝑄𝑙. (121)