Abstract. We calculate the transfinite diameter for the real unit ball, B

THE TRANSFINITE DIAMETER OF THE REAL BALL AND SIMPLEX

T. BLOOM*, L. BOS, AND N. LEVENBERG

Abstract. We calculate the transfinite diameter for the real unit ball, B

d

:= {x ∈ R

^d

: |x| ≤ 1} and the real unit simplex, T

d

:= {x ∈ R

^d+

: P

d

j=1

x

_j

≤ 1}.

1. Introduction

Suppose that K ⊂ C

^d

is compact. The transfinite diameter (and the associated notion of capacity) of K is a measure of the size of K, important in classical Complex Potential Theory (for d = 1) and Pluripotential Thoery (for d > 1). It is defined as follows. For n ∈ Z

+

consider the monomials of degree at most n, z

^γ

, |γ| ≤ n which we order as {e

1

(z), e

2

(z), · · · , e

mn

(z)}

so that the ordering respects the degree, i.e., |γ

_j

| < |γ

_i

| implies that j <

i, where e

_i

(z) = z

^γi

. Here m

_n

:=

^n+d_n

is the dimension of the space of polynomials of degree at most n in d complex variables. Then for m

_n

points z

_j

∈ K, 1 ≤ j ≤ m

_n

, the Vandermonde determinant of degree n is defined to be

vdm(z

1

, z

2

, · · · , z

mn

) := det([e

i

(z

j

)])

1≤i,j≤mn

).

The transfinite diameter of K is δ(K) := lim

n→∞



z1,··· ,z

max

mn∈K

|vdm(z

1

, z

2

, · · · , z

mn

)|



1/`n

where `

_n

:=

_d+1^d

nm

_n

is the degree of homogeneity of vdm(z

₁

, z

₂

, · · · , z

_mn

) considered as a polynomial on K

^mn

.

That the limit exists is a result of Zaharjuta [Z] where the following remarkable formula is proved:

(1.1) δ(K) = exp

 1

vol(S

d

) Z

Sd

log(τ (θ))dV



where

(1.2) S

_d

:= {θ ∈ R

^d

: θ

_j

≥ 0, 1 ≤ j ≤ d, and

d

X

j=1

θ

_j

= 1}

2010 Mathematics Subject Classification. Primary 32U15.

Key words and phrases. Transfinite Diameter, Real Ball, Real Simplex.

*Supported in part by an NSERC of Canada grant.

1

is a symmetric simplex (of dimension d − 1) in R

^d

and, for θ ∈ S

_d

, τ (θ) = τ (θ, K) := lim sup

i→∞, αi/|αi|→θ

inf{ke

_i

(z) + X

j<i

c

_j

e

_j

(z)k

^1/|α_K ^i|

}

is the so-called directional Chebyshev constant. We note that, for θ ∈ int(S

_d

), Zaharjuta showed that the lim sup in the definition of τ (θ) may be replaced by an ordinary limit.

In certain cases explicit formulas for the transfinite diameter have been calculated. J¸edrzejowski [J] has given the following formula for the unit complex euclidean ball, B

d

:= {z ∈ C

^d

: |z| ≤ 1} :

(1.3) δ(B

^d

) = exp − 1

2

d

X

j=2

1 j

! .

From the product formula of Schiffer and Siciak [SS] we also have, for the unit cube Q

_d

:= [−1, 1]

^d

,

(1.4) δ(Q

_d

) = 1

2 .

Rumely [R] has given a beautiful integral formula for the transfinite diameter using notions of Pluripotential theory (see also [DR]). However, this appears to be difficult to explicitly evaluate, at least in the case of the real ball in which we are interested. In this paper we will use Zaharjuta’s formula (2.9) directly, by finding a formula for τ (θ) and then computing its integral over the simplex, to give a compact formula (Theorem 2.5 below) for the transfinite diameter of the real euclidean unit ball B

_d

:= {x ∈ R

^d

:

|x| ≤ 1} and the real unit simplex, T

_d

:= {x ∈ R

^d+

: P

d

j=1

x

_j

≤ 1}.

2. The Ball

We begin with some standard facts about univariate orthogonal polyno- mials.

2.1. Univariate Gegenbauer Polynomials. Let C

_n^λ

(x), n ∈ Z

+

, λ >

−1/2, denote the Gegenbauer polynomials of degree n and parameter λ, i.e., the classical univariate polynomials orthogonal on [−1, 1] with respect to the inner product

(2.1) hf, gi

_λ

:=

Z

1

−1

f (x)g(x)w

_λ

(x)dx where

w

_λ

(x) := (1 − x

²

)

^λ−1/2

. It is known that (see e.g. [AS])

C

_n^λ

(x) = k

_n

x

ⁿ

+ lower degree terms

where

(2.2) k

_n

:= 2

ⁿ

Γ(λ + n)

n!Γ(λ) and that

Z

1

−1

(C

_n^λ

(x))

²

w

_λ

(x)dx = π2

^1−2λ

Γ(n + 2λ) n!(n + λ)(Γ(λ))

²

. If we let b C

_n^λ

(x) :=

_k¹

n

C

_n^λ

(x) denote the associated monic orthogonal poly- nomials, it follows easily that

C b

_n^λ

(x) = x

ⁿ

+ lower degree terms and that

h(n, λ) :=

Z

1

−1

( b C

_n^λ

(x))

²

w

_λ

(x)dx

= π2

^1−2(n+λ)

Γ(n + 1)Γ(n + 2λ) (Γ(n + λ))

²

(n + λ) . (2.3)

We will make use of the following the nth root asymptotics of h(n, λ).

Lemma 2.1. Suppose that a

_n

, b

_n

≥ 0 are such that lim

_n→∞

a

_n

/n = a and lim

_n→∞

b

_n

/n = b with a + b > 0. Then

n→∞

lim h(a

_n

, b

_n

)

^1/n

= a

^a

(a + 2b)

^a+2b

2

^2(a+b)

(a + b)

^2(a+b)

. Proof. From (2.3) we have

h(a

_n

, b

_n

) = π2

^1−2(an+bn)

Γ(a

n

+ 1)Γ(a

n

+ 2b

n

) (Γ(a

_n

+ b

_n

))

²

(a

_n

+ b

_n

) .

Clearly, the terms π

^1/n

and (a

_n

+ b

_n

)

^1/n

both tend to 1 and hence can be ignored. Further,

2

^1−2(an+bn)

1/n

= 2

^{1/n−2(an/n+bn/n)}

→ 2

^−2(a+b)

and hence we are left with showing that

n→∞

lim

 Γ(a

_n

+ 1)Γ(a

_n

+ 2b

_n

) (Γ(a

_n

+ b

_n

))

²



1/n

= a

^a

(a + 2b)

^a+2b

(a + b)

^2(a+b)

.

But this is an easy consequence of Stirling’s formula, and we are done.  2.2. A Family of Orthogonal Polynomials on B

_d

. For µ > −1/2 and x = (x

₁

, x

₂

, · · · , x

_d

) ∈ B

_d

let

W

_µ

(x) = (1 − |x|

²

)

^µ−1/2

be a generalized Gegenbauer weight. Now, suppose that α ∈ Z

^d+

is a multi- index. We define (cf. Prop. 2.3.2 of [DX]) the polynomials

P

_α

(x) = b C

_α^λ1₁

(x

₁

) × (1 − x

²₁

)

^α2/2

C b

_α^λ2₂

( x

2

p1 − x

²₁

)

!

× (1 − x

²₁

− x

²₂

)

^α3/2

C b

_α^λ3

3

( x

₃

p1 − x

²₁

− x

²₂

)

!

× · · · ×

×



(1 − x

²₁

− · · · − x

²_d−1

)

^αd/2

C b

_α^λd

d

( x

_d

q

1 − x

²₁

− · · · − x

²_d−1

)



 where

λ

₁

= µ + α

₂

+ α

₃

+ · · · + α

_d

+ (d − 1)/2, λ

₂

= µ + α

₃

+ α

₄

+ · · · + α

_d

+ (d − 2)/2, λ

₃

= µ + α

₄

+ α

₅

+ · · · + α

_d

+ (d − 3)/2,

·

· λ

d

= µ, that is

λ

_j

:= µ + (d − j)/2 +

d

X

k=j+1

α

_k

.

Note that since the weight w

_λ

(x) is symmetric, the polynomials b C

_n^λ

(x) are even for n even and odd for n odd. Hence the P

_α

, as defined above, are indeed d-variate polynomials.

Lemma 2.2. The polynomials P

_α

(x) are orthogonal with respect to the inner product

(2.4) hf, gi

_µ

:=

Z

Bd

f (x)g(x)W

_µ

(x)dx.

Proof. This is given in Prop. 2.3.2 of [DX]. 

Lemma 2.3. The polynomials P

_α

(x) are monic, i.e., of the form P

_α

(x) = x

^α

+ lower order terms

where the ordering of the monomials is graded-lexicographic with x

₁

≺ x

₂

≺

· · · ≺ x

_d

.

Moreover,

Z

Bd

(P

α

(x))

²

W

µ

(x)dx =

d

Y

j=1

h(α

j

, λ

j

).

Proof. The mononoticity is obvious from the construction. The norm is easily calculated by writing

Z

Bd

f (x)W

_µ

(x)dx = Z

Bd−1

Z

√

1−|x⁰|²

−

√

1−|x⁰|²

f (x

⁰

, x

_d

)W

_µ

(x

⁰

, x

_d

)dx

_d

dx

⁰

, where x

⁰

:= (x

₁

, · · · , x

_d−1

), substituting y

_d

:= x

_d

/p1 − |x

⁰

|

²

, and proceed-

ing by induction. 

2.2.1. The Measures W

_µ

dx are Bernstein-Markov. A measure ν on a com- pact set K ⊂ C

^d

is said to be a Bernstein-Markov measure when there exists constants C(n) with the property that

n→∞

lim C(n)

^1/n

= 1 such that for all polynomials p of degree n,

kpk

_K

≤ C(n)kpk

_L2(K;ν)

.

For many purposes, including the calculation of transfinite diameters, this means that the uniform norm can be substituted by the L

₂

norm with respect to the measure ν.

It turns out that, for the measures W

_µ

(x)dx, µ ≥ 0, there is a stronger statement. Specifically, from Lemma 1 of [B2], it follows that, for α = 0, (2.5) kpk

_Bd

≤

s 2 ω

d

n + d d



+ n + d − 1 d



kpk

_L2(Bd;W0dx)

for all polynomials p of degree n. Here ω

_d

is the surface area of the d dimen- sional unit sphere in R

^d+1

. In other words, for this measure C(n) = O(n

^d/2

) is of polynomial growth. Similarly, the considerations of [B2] show that C(n) is also of polynomial growth for all µ ≥ 0.

2.3. The Directional Chebyshev Constants. Since the measures W

_µ

(x)dx, µ ≥ 0 are Bernstein-Markov measures, we may use any of the associated 2-norms,

kf k

µ

:= hf, f i

^1/2_µ

,

to compute the directional Chebyshev constants (cf. [B1, p. 320]). Specifi-

cally consider θ ∈ S

_d

, the unit simplex in R

^d

(see (1.2)). Then for n = |α| =

α

₁

+ · · · α

_d

τ (θ) = lim

α/n→θ

kP

_α

k

^1/n_µ

= lim

α/n→θ

(

_d

Y

j=1

h(α

_j

, λ

_j

) )

_2n¹

by Lemma 2.3.

Suppose now that θ ∈ S

d

with θ > 0 (i.e., each component is positive).

Then under the hypothesis that α/n → θ we have

n→∞

lim α

_j

/n = θ

_j

, 1 ≤ j ≤ d, (2.6)

n→∞

lim λ

_j

/n =

d

X

k=j+1

θ

_k

, 1 ≤ j ≤ d.

(2.7)

Hence, by Lemma 2.1

n→∞

lim h(α

_j

, λ

_j

)

^1/n

= θ

^θ_j^j

(θ

_j

+ 2 P

d

k=j+1

θ

_k

)

^{θj+2Pdk=j+1θk}

2

^2(Pdk=jθk)

( P

d

k=j

θ

_k

)

^2(Pdk=jθk)

=: H

_j

(θ).

(2.8)

We thus may conclude

Lemma 2.4. For θ ∈ S

_d

with θ > 0, τ (θ) =

(

_d

Y

j=1

H

_j

(θ) )

1/2

.

2.4. Zaharjuta’s Formula for The Transfinite Diameter. We will use Zaharjuta’s formula for the transfinite diameter, given in the introduction:

(2.9) δ(B

_d

) = exp

 1

vol(S

_d

) Z

Sd

log(τ (θ))dV

 . Theorem 2.5. The transfinite diameter of the unit ball B

_d

is:

(a) for d even, δ(B

_d

) = 1

2 exp − 1 4

2d + 1 d

d

X

j=1

1 j + 1

2 + 1

2 log(2) + 1 4d

d

X

j=1

(−1)

^j

j

! , (b) for d odd,

δ(B

_d

) = 1

2 exp − 1 4

2d + 1 d

d

X

j=1

1 j + 1

2 + d − 1

2d log(2) − 1 4d

d

X

j=1

(−1)

^j

j

! . Remark. For d = 1 the formula reduces to the classical δ([−1, 1]) = 1/2 and for d = 2 we obtain δ(B

₂

) = 1/ √

2e, in agreement with the result of

[B1].

Proof. We use the Zaharjuta formula together with the formula for τ (θ) given in Lemma 2.1 to obtain

δ(B

_d

) = exp  (d − 1)!

2 (C

_d

+ F

_d

− D

_d

− E

_d

)



where

C

_d

=

d

X

j=1

Z

Sd

log(θ

^θ_j^j

)dV = − 1 (d − 1)!

d

X

j=2

1 j ,

D

_d

=

d

X

j=1

Z

S_d

log 

2

^2Pdk=jθk



dV = d + 1

(d − 1)! log(2),

E

_d

=

d

X

j=1

Z

S_d

log





d

X

k=j

θ

_k

!

²

(

^Pdk=jθ_k

)

 dV = − 1 2(d − 2)! , and

F

_d

=

d

X

j=1

Z

Sd

log



 θ

_j

+ 2

d

X

k=j+1

θ

_k

!(

^θj+2Pdk=j+1θk

)

 dV

= 1 2

1 (d − 2)!



2 log(2) − 1 + Z

2

1

(x − 1)

^d−2

x log(x) dx



− 1 2

1 d!

d

X

j=2

1 j . The values for the integrals are given in a sequence of lemmas below. Note that vol(S

_d

) = 1/(d − 1)!; putting them together and simplifying gives the

result. 

We can express integrals over the simplex S

_d

as univariate integrals by means of B-splines.

Lemma 2.6. Suppose that a

₁

≤ a

₂

≤ · · · ≤ a

_d

with d ≥ 2 and that f ∈ L

₁

[a

₁

, a

_d

]. Then

Z

S_d

f (

d

X

j=1

θ

_j

a

_j

)dV = 1

(a

_d

− a

₁

)(d − 2)!

Z

∞

−∞

f (x)B(x | a

₁

, a

₂

, · · · a

_d

)dx where

B(x | a

₁

, a

₂

, · · · a

_d

) := (a

_d

− a

₁

)(· − x)

^d−2₊

[a

₁

, a

₂

, · · · , a

_d

] is the B-spline of degree d − 2 and knots a

₁

, a

₂

, · · · , a

_d

. Here

y

₊

:=  y y ≥ 0 0 y < 0

and f [a

₁

, a

₂

, · · · , a

_d

] denotes the divided difference of the function f at the

points a

_i

.

Proof. This is a standard formula of spline theory, based on the fact that B-splines are the Peano kernel for divided differences; see, for example, [deB,

p. 88]. 

We will need the following fact.

Lemma 2.7. For 1 < j ≤ d and d ≥ 2, we have B(x | 0, · · · , 0

| {z }

j−1

, 1, · · · , 1

| {z }

d−j+1

) =



_d−2

j−2

x

^d−j

(1 − x)

^j−2

, x ∈ [0, 1]

0, otherwise .

Proof. This is again a standard fact that follows easily from the recurrence formula for B-splines (see again [deB, p. 89]).  Lemma 2.8. For 1 ≤ j ≤ d, we have

Z

Sd

θ

_j

log(θ

_j

)dV = 1 (d − 2)!

Z

1 0

(1 − x)

^d−2

x log(x)dx

= − 1 d!

d

X

j=2

1 j . Hence

C

_d

=

d

X

j=1

Z

Sd

θ

_j

log(θ

_j

)dV = − 1 (d − 1)!

d

X

j=2

1 j .

Proof. By symmetry me need only consider j = d. Then, using the integral formula of Lemma 2.6, we have

Z

Sd

θ

d

log(θ

d

)dV = 1

(1 − 0)(d − 2)!

Z

∞

−∞

x log(x)B(x | 0, 0, · · · , 0

| {z }

d−1

, 1)dx

= 1

(d − 2)!

Z

1 0

x log(x)(1 − x)

^d−2

dx since, by Lemma 2.7 with j = d − 1,

B(x | 0, · · · , 0

| {z }

d−1

, 1) =  (1 − x)

^d−2

, x ∈ [0, 1]

0, otherwise . By Lemma 2.9 with m = d − 2 we obtain

1 (d − 2)!

(

− 1

(d − 1)d

d

X

j=2

1 j

)

and the result follows. 

Lemma 2.9. For m ∈ Z

+

we have Z

1

0

(1 − x)

^m

x log(x)dx = − 1

(m + 1)(m + 2)

m+2

X

j=2

1

j .

Proof. This is a special case of Formula 1. of §4.253 of [GR], however, for completeness, we provide an elementary proof. Let A

m

denote the integral in question. Then integrating

A

_m+1

= Z

1

0

(1 − x)

^m+1

x log(x)dx.

by parts with u = (1 − x)

^m+1

and v

⁰

= x log(x) we easily obtain the recur- rence

A

_m+1

= m + 1

m + 3 A

_m

− 1

(m + 2)(m + 3)

²

from which it is easy to verify the stated formula by induction.  Lemma 2.10. For d ≥ 2 we have

D

_d

=

d

X

j=1

Z

S_d

log 

2

^2Pdk=jθk



dV = d + 1

(d − 1)! log(2).

Proof. The j = 1 term is slightly different. In fact, for j = 1, P

d

k=j

θ

k

= 1 and the integrand is just 2 log(2) and the integral

(2.10)

Z

Sd

log 

2

^2Pdk=1θk



dV = 2 log(2)vol(S

_d

) = 2 log(2) (d − 1)! . For the other terms we compute

d

X

j=2

Z

Sd

log 

2

^2Pdk=jθk

 dV

= 2 log(2)

d

X

j=2

Z

Sd

d

X

k=j

θ

_k

! dV

= 2 log(2)

d

X

j=2

1

(1 − 0)(d − 2)!

Z

1 0

xB(x | 0, · · · , 0

| {z }

j−1

, 1, · · · , 1

| {z }

d−j+1

)dx

= 2 log(2) (d − 2)!

d

X

j=2

Z

1 0

x d − 2 j − 2



x

^d−j

(1 − x)

^j−2

dx (by Lemma 2.7)

= 2 log(2) (d − 2)!

Z

1 0

x

d

X

j=2

d − 2 j − 2



x

^d−j

(1 − x)

^j−2

dx

= 2 log(2) (d − 2)!

Z

1 0

x

d−2

X

j=0

d − 2 j



x

^(d−2)−j

(1 − x)

^j

dx

= 2 log(2) (d − 2)!

Z

1 0

x dx = log(2)

(d − 2)! .

Combining this with (2.10) gives

d

X

j=1

Z

Sd

log 

2

^2Pdk=jθk



dV = 2 log(2)

(d − 1)! + log(2) (d − 2)!

and the result follows.

As pointed out by an anonymous reviewer, this integral can also be computed by completely elementary means. Specifically, first observe that by symmetry,

Z

Sd

θ

_j

dV = Z

Sd

θ

_k

dV = 1

d vol(S

_d

) = 1 d! . Then

D

_d

=

d

X

j=1

Z

Sd

log 

2

^2Pdk=jθk



dV = 2 log(2)

d

X

j=1 d

X

k=j

1 d!

= 2

d! log(2)

d

X

j=1

(d − j + 1) = 2

d! log(2) d(d + 1) 2

= d + 1

(d − 1)! log(2).

 Lemma 2.11. For d ≥ 2 we have

E

_d

=

d

X

j=1

Z

Sd

2

d

X

k=j

θ

_k

! log

d

X

k=j

θ

_k

!

dV = − 1 2(d − 2)! . Proof. The j = 1 term is 0 since P

d

k=1

θ

_k

= 1 and so it can be ignored.

Hence we compute

d

X

j=2

Z

Sd

2

d

X

k=j

θ

_k

! log

d

X

k=j

θ

_k

! dV

= 2

d

X

j=2

Z

Sd

d

X

k=j

θ

_k

! log

d

X

k=j

θ

_k

! dV

= 2

d

X

j=2

1 (1 − 0)(d − 2)!

Z

1 0

x log(x)B(x | 0, · · · , 0

| {z }

j−1

, 1, · · · , 1

| {z }

d−j+1

)dx

= 2

(d − 2)!

Z

1 0

x log(x)

d

X

j=2

B(x | 0, · · · , 0

| {z }

j−1

, 1, · · · , 1

| {z }

d−j+1

)dx

= 2

(d − 2)!

Z

1 0

x log(x) dx (as in Lemma 2.10)

= 2

(d − 2)!



− 1 4



and the result follows.  Lemma 2.12. For d ≥ 2 we have

F

_d

=

d

X

j=1

Z

Sd

θ

_j

+ 2

d

X

k=j+1

θ

_k

!

log θ

_j

+ 2

d

X

k=j+1

θ

_k

! dV

= 1 2

1 (d − 2)!



2 log(2) − 1 + Z

2

1

(x − 1)

^d−2

x log(x) dx



− 1 2

1 d!

d

X

j=2

1 j .

The univariate integral above is evaluated in Lemma 2.13.

Proof. Let I

_j

:=

Z

Sd

θ

_j

+ 2

d

X

k=j+1

θ

_k

!

log θ

_j

+ 2

d

X

k=j+1

θ

_k

! dV.

We must compute P

d

j=1

I

_j

. First note that, by Lemma 2.6, I

₁

=

Z

Sd

θ

₁

+ 2

d

X

k=2

θ

_k

!

log θ

₂

+ 2

d

X

k=2

θ

_k

! dV

= 1

2 − 1 1 (d − 2)!

Z

∞

−∞

x log(x)B(x | 1, 2, · · · , 2

| {z }

d−1

)dx

= 1

(d − 2)!

Z

∞

−∞

x log(x)B(x | 1, 2, · · · , 2

| {z }

d−1

)dx, (2.11)

while

I

_d

= Z

Sd

(θ

_d

) log(θ

_d

)dV

= 1

1 − 0 1 (d − 2)!

Z

∞

−∞

x log(x)B(x | 0, · · · , 0

| {z }

d−1

, 1)dx

= 1

(d − 2)!

Z

∞

−∞

x log(x)B(x | 0, · · · , 0

| {z }

d−1

, 1)dx

Further, for 2 ≤ j ≤ (d − 1), I

_j

=

Z

Sd

θ

_j

+ 2

d

X

k=j+1

θ

_k

!

log θ

_j

+ 2

d

X

k=j+1

θ

_k

! dV

= 1

2 − 0 1 (d − 2)!

Z

∞

−∞

x log(x)B(x | 0, · · · , 0

| {z }

j−1

, 1, 2, · · · , 2

| {z }

d−j

)dx

= 1 2

1 (d − 2)!

Z

∞

−∞

x log(x)B(x | 0, · · · , 0

| {z }

j−1

, 1, 2, · · · , 2

| {z }

d−j

)dx.

Hence 1 2 I

1

+

d−1

X

j=2

I

j

+ 1 2 I

d

= 1 2

1 (d − 2)!

Z

∞

−∞

x log(x)







d

X

j=1

B(x | 0, · · · , 0

| {z }

j−1

, 1, 2, · · · , 2

| {z }

d−j

)





 dx.

However, just as Lemma 2.10, it can be shown (cf. [deB]) that

d

X

j=1

B(x | 0, · · · , 0

| {z }

j−1

, 1, 2, · · · , 2

| {z }

d−j

) ≡ 1

on its support, i.e., on (0, 2) in this case. Thus 1

2 I

₁

+

d−1

X

j=2

I

_j

+ 1

2 I

_d

= 1 2

1 (d − 2)!

Z

2 0

x log(x)dx

= 1 2

1

(d − 2)! (2 log(2) − 1).

It follows that

d

X

j=1

I

_j

= 1 2

1

(d − 2)! (2 log(2) − 1) + 1

2 (I

₁

+ I

_d

).

But, by (2.11),

I

₁

= 1 (d − 2)!

Z

∞

−∞

x log(x)B(x | 1, 2, · · · , 2

| {z }

d−1

)dx

= 1

(d − 2)!

Z

2 1

(x − 1)

^d−2

x log(x)dx and from Lemma 2.8,

I

_d

= − 1 d!

d

X

j=2

1 j .

Putting these together yields the result. 

Lemma 2.13. For m ∈ Z

+

we have Z

2

1

(x − 1)

^m

x log(x)dx = a

_m

log(2) + b

_m

where

a

_m

:=



 

 

 

  2

m + 1 m even 2

m + 2 m odd

and

b

_m

:= − (−1)

^m

(m + 1)(m + 2)

(

(−1)

^m

−

m+2

X

k=1

(−1)

^k

/k )

.

Proof. Let A

_m

denote the integral in question. Just as before we may integrate by parts (with u = (x − 1)

^m

and v

⁰

= x log(x)) to show that

A

_m+1

= − m + 1

m + 3 A

_m

+ 4

m + 3 log(2) − 2m + 5 (m + 2)(m + 3)

²

and from this one can easily verify the given formula by induction.  3. The Simplex

By the mapping result of Bloom and Calvi [BC] we immediately obtain Theorem 3.1. The transfinite diameter of the simplex T

d

is given by

δ(T

_d

) = (δ(B

_d

))

²

.

Proof. Just note that T

_d

is the image of B

_d

under the mapping (x

₁

, x

₂

, · · · , x

_d

) 7→ (x

²₁

, x

²₂

, · · · , x

²_d

).

 4. Concluding Remarks

Using our formulas for δ(B

_d

) and the formula (1.3) for δ(B

d

) we easily find that

d→∞

lim δ(B

d

) δ(B

_d

) = √

2

showing that the real and complex balls have finitely comparable transfinite diameters.

Further, for the inscribed cube 1

√ d Q

d

⊂ B

d

we have

lim

d→∞

δ(B

_d

) δ(

^√1

d

Q

_d

) = p

2 exp(1 − γ)

where γ is Euler’s constant, while for the superscribed cube,

d→∞

lim δ(B

d

) δ(Q

_d

) = 0.

Since 1

√ d Q

_d

⊂ B

_d

⊂ Q

_d

it is natural to ask why δ(B

_d

) is comparable to the inscribed cube and not to the superscribed cube. However, even in the euclidean case,

vol(B

_d

) = π

^d/2

/Γ(1 + d/2) ∼ 1

√ πd

r 2πe d

!

d

,

by Stirling’s formula, showing that the volume of B

_d

behaves like that of a cube with side length proportional to 1/ √

d, i.e., like that of the inscribed cube.

Acknowledgments. We would like to thank the anonymous reviewers for their careful reading of the manuscript and for their perceptive comments.

References

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[B1] L. Bos, A characteristic of points in R

²

having Lebesgue function of polynomial growth, J. Approx. Theory, 56 (1989), 316–329.

[B2] L. Bos, Asymptotics for the Christoffel function for Jacobi like weights on a ball in R

^m

, New Zealand J. Math., 23 (1994), 99 – 109.

[BC] T. Bloom and J.-P. Calvi, On the multivariate transfinite diameter, Ann. Polon.

Math. 72 (1999), 285-305.

[deB] C. de Boor, A Practical Guide to Splines, Revised Edition, Springer, 2001.

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Math. 611 (2007), 145 – 161.

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[GR] I. Gradshteyn and I. Ryhzik, Tables of Integrals, Series, and Products, Seventh Edition, Academic Press, 2007.

[J] M. J¸ edrzejowski, The homogeneous transfinite diameter of a compact subset of C

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[SS] M. Schiffer and J. Siciak, Transfinite diameter and the analytic continuation of functions of two complex variables, in Studies in Mathematical Analysis and Re- lated Topics, pp. 341 – 358, Stanford Univ. Press, 1962.

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ⁿ

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University of Toronto, Toronto, Ontario M5S 2E4 Canada E-mail address: bloom@math.toronto.edu

University of Verona, Verona, Italy E-mail address: leonardpeter.bos@univr.it

Indiana University, Bloomington, IN 47405 USA

E-mail address: nlevenbe@indiana.edu