• Non ci sono risultati.

Since the origin of E is in the convex hull of {x1


Academic year: 2021

Condividi "Since the origin of E is in the convex hull of {x1"


Testo completo


Problem 11825

(American Mathematical Monthly, Vol.122, March 2015) Proposed by M. Dinc˘a and S. Radulescu (Romania).

Let E be a normed linear space. Given x1, . . . xn∈ E (with n ≥ 2) such that kxkk = 1 for 1 ≤ k ≤ n and the origin of E is in the convex hull of {x1, . . . , xn}, prove that kx1+ · · · + xnk ≤ n − 2.

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

Since the origin of E is in the convex hull of {x1, . . . , xn}, it follows that there exist non-negative numbers α1, . . . , αn such that

α1+ · · · + αn = 1 and α1x1+ · · · + αnxn = 0.

We show that αk∈ [0, 1/2] for 1 ≤ k ≤ n. In fact

α1= k − α1x1k = kα2x2+ · · · + αnxnk ≤ α2kx2k + · · · + αnkxnk = α2+ · · · + αn= 1 − α1. which implies that α1≤ 1/2. The other cases are similar. Hence

kx1+ · · · + xnk = kx1+ · · · + xn− 2(α1x1+ · · · + αnxn)k = k(1 − 2α1)x1+ · · · + (1 − 2αn)xnk

≤ (1 − 2α1)kx1k + · · · + (1 − 2αn)kxnk = (1 − 2α1) + · · · + (1 − 2αn) = n − 2.

Finally, we note that right-hand side of the inequality can not be lowered. Take any unit vector x1, let x2= · · · = xn−1= x1and xn = −x1. Then the origin of E is in the convex hull of {x1, . . . , xn} because 12x1+12xn = 0, and kx1+ · · · + xnk = k(n − 2)x1k = n − 2.


Documenti correlati

Our detailed spectroscopic observations of RR Lyrae, obtained over an entire Blazhko period, make it more difficult to explain the Blazhko effect with the current resonance mod-

Fibra di carbonio T700S & HFM monoscocca Finiture: UD, 3K, verniciato Columbus Forcellini in carbonio - inserti in alluminio Sterzo conico 1-1/8” - 1-1/2”.. Lunghezza sterzo:

the field occupied by this parameter might be utilized to communicate the required advance to the mobile for the frame synchronization. Therefore, 7 b are sufficient to

Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to 3. 66 Departamento de Fisica, Universidad Nacional de Colombia, Bogota,

We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative

consequently they will have been liable to modification; and this we find has been the case; for if we compare the present Alpine plants and animals of the several great

Dopo la battuta di arresto del 1999 la prima occasione di rilievo per vedere riconosciuto il patrimonio linguistico di rom e sinti come parte della storia e della cultura

We recall that a CAT (constant amortized time) algorithm for the exhaustive generation of parallelogram polyominoes of size n has been recently proposed in [15], where it is shown

La critica e la teoria della letteratura contemporanee a Tolstoj si erano concentrate principalmente sui contenuti della sua opera, giudicandoli o spiegandoli in

La comunicazione economico-finanziaria d’impresa: disclosure obbligatoria e disclosure volontaria 2.1 INTRODUZIONE: profili evolutivi……….………...97.. 2.2 Tipologie

Now that all the ingredients are in place, we can simply remind readers of one of the most fundamental results of analysis: Brouwer’s (1910) theorem, whereby each continuous mapping

decreases the activation energy of the forward reaction and increases the activation energy of the reverse reaction.. increases the activation energy of the forward reaction

We prove here that collapsing only occurs if the mean curvature/pressure of the bulky regions is negative, and that, when this last property holds, the whole soap film lies in

Impact between rigid bodies (*) (*) having a rotation, an angular vel, etc.  So far, we only considered collision between particles (no rotation, no

If the overlapping rate between fragments is adequate and these fragments are long enough to encompass multiple variant sites (as expected by sequence technology improvement), a

C syndrome (Opitz trigonocephaly syndrome, OMIM 211750), presumably an autosomal recessive disorder, is char- acterized by trigonocephaly, unusual facies (up- slanting

Like P53, mutations in RB or its associated tumor suppressor genes occur frequently, and disabling this pathway may be required for the formation of human cancer cells (81,82)..

For larger defects, it may not be possible to close the fascia, but there may be enough skin to achieve skin closure over the viscera.. This technique was original- ly described

In summary, thanks to a simple Monte Carlo study we have shown that groomed event- shapes in e + e − collisions at LEP energies, such as soft-drop thrust, are characterised by

Per ricavare dalle funzioni di correlazione della carica topologica misurate su reticolo il loro limite continuo ci sono diversi metodi [13]: le costanti di rinormalizzazione

Torna- ta alla ribalta nelle molte forme di neo-razzismo biologico (Bonilla- Silva), così come nei razzismi culturalisti, dove un concetto di cultura impermeabile e reificata prende

Abstract: Aim of this study was to evaluate the plasma ratio between L -tryptophan (TRP) and five large neutral amino acids (isoleucine + leucine + phenylalanine + tyrosine +

A social or political conflict, whether it is a non violent or a violent one, is always a complex system, where complexity stems from many different and sometimes unrelated