On average distances and the geometry of Banach spaces

Baronti, Marco; Casini, Emanuele; Papini, Pier Luigi

doi:10.1016/s0362-546x(99)00112-1

Cited by 10 publications

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“…On the other hand for each 1 < p < ∞ we construct a nonreflexive Banach space Y p whose unit ball does not contain a sequence with the separation constant greater than 2 1/p such that its convex hull is arbitrarily close to the unit sphere. These results provide answers to two of the questions posed in [2]. Let us however mention that our notation does not coincide with that of [2].…”

Section: Introductionmentioning

confidence: 83%

“…These results provide answers to two of the questions posed in [2]. Let us however mention that our notation does not coincide with that of [2].…”

Section: Introductionmentioning

confidence: 83%

“…They are strongly related to the problem of packing balls (see [15]), average distances (see [2]) and infinite dimensional convexity (see [9]). The last notion in turn plays an important role in metric fixed point theory (see [8] and [1]).…”

Section: Introductionmentioning

confidence: 99%

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Separated sequences in nonreflexive Banach spaces

Kryczka

2000

Proc. Amer. Math. Soc.

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Abstract. We prove that there is c > 1 such that the unit ball of any nonreflexive Banach space contains a c-separated sequence. The supremum of these constants c is estimated from below by 5 √ 4 and from above approximately by 1.71. Given any p > 1, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed 2

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Section: Introductionmentioning

confidence: 83%

“…These results provide answers to two of the questions posed in [2]. Let us however mention that our notation does not coincide with that of [2].…”

Section: Introductionmentioning

confidence: 83%

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Separated sequences in nonreflexive Banach spaces

Kryczka

2000

Proc. Amer. Math. Soc.

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“…In particular, calculation of rendezvous numbers of unit spheres in Banach spaces fascinated many authors. (See, e.g., Baronti, Casini, Papini [2], García-Vázquez, Villa [11], Lin [13], Wolf [21]). …”

Section: Introductionmentioning

confidence: 99%

Rendezvous numbers of metric spaces – a potential theoretic approach

Farkas

Révész

2006

Arch. Math.

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Abstract. The present work draws on the understanding how notions of general potential theory -as set up, e.g., by Fuglede -explain existence and some basic results on the "magical" rendezvous numbers. We aim at a fairly general description of rendezvous numbers in a metric space by using systematically the potential theoretic approach.In particular, we generalize and explain results on invariant measures, hypermetric spaces and maximal energy measures, when showing how more general proofs can be found to them. IntroductionIt was proved by O. Gross that for a compact, connected metric space (X, d) there exists a unique number r = r(X) such that for every finite point system x 1 , . . . , x n ∈ X, n ∈ N one always finds an x ∈ X with Our aim is to put the investigations on the existence and uniqueness of rendezvous numbers in the framework of abstract potential theory, which has been around since the 60s, but apparently has not gained its due recognition in this field. In this paper we continue [6] First we spend some words on technicalities and recall the appropriate setting of potential theory in locally compact spaces. For convenience we add +∞ to the set of real numbers, i.e.,2000 Mathematics Subject Classification. Primary: 31C15. Secondary: 28A12, 54D45. Key words and phrases. Locally compact Hausdorff topological spaces, potential theoretic kernel function in the sense of Fuglede, potential of a measure, energy integral, energy and capacity of a set, Chebyshev constant, (weak) rendezvous number, average distance, minimax theorem, invariant measure, positive definite kernel, maximum principle, Frostman's equilibrium theorem.

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“…DEFINITION 2: Let H,L C X be fixed, and n € VJli{X) be arbitrary. First put (2) Q(n,H) := supU"(x) , and also Q(fJ.,H) := inf U"{x) .…”

Section: Introductionmentioning

confidence: 99%

Rendezvous numbers in normed spaces

Farkas¹,

Révész²

2005

Bull. Austral. Math. Soc.

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In previous papers, we used abstract potential theory, as developed by Puglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We considered Chebyshev constants and energies as two variable set functions, and introduced a modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces.Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these magical numbers to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of L p spaces is obtained for all p > 0.

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On average distances and the geometry of Banach spaces

Cited by 10 publications

References 0 publications

Separated sequences in nonreflexive Banach spaces

Separated sequences in nonreflexive Banach spaces

Rendezvous numbers of metric spaces – a potential theoretic approach

Rendezvous numbers in normed spaces

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