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A new type of orthogonality for normed planes

Horst Martini, Margarita Spirova (2010)

Czechoslovak Mathematical Journal

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d 3 .

A selection theorem of Helly type and its applications

Ehrhard Behrends, Kazimierz Nikodem (1995)

Studia Mathematica

We prove an abstract selection theorem for set-valued mappings with compact convex values in a normed space. Some special cases of this result as well as its applications to separation theory and Hyers-Ulam stability of affine functions are also given.

Between Paouris concentration inequality and variance conjecture

B. Fleury (2010)

Annales de l'I.H.P. Probabilités et statistiques

We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.

Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov (2007)

Colloquium Mathematicae

Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by R B ( T ) the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science R B ( T ) is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein). Using methods...

Estimates on inner and outer radii of unit balls in normed spaces

Horst Martini, Zokhrab Mustafaev (2011)

Colloquium Mathematicae

The purpose of this paper is to continue the investigations on extremal values for inner and outer radii of the unit ball of a finite-dimensional real Banach space for the Holmes-Thompson and Busemann measures. Furthermore, we give a related new characterization of ellipsoids in d via codimensional cross-section measures.

Euclidean arrangements in Banach spaces

Daniel J. Fresen (2015)

Studia Mathematica

We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Kashin decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's position with a Euclidean ball of a given radius, which leads to a new and simplified proof of the randomized isomorphic Dvoretzky theorem. Our results also include a characterization of spaces with nontrivial cotype in terms of arrangements of Euclidean subspaces.

Extremal sections of complex l p -balls, 0 < p ≤ 2

Alexander Koldobsky, Marisa Zymonopoulou (2003)

Studia Mathematica

We study the extremal volume of central hyperplane sections of complex n-dimensional l p -balls with 0 < p ≤ 2. We show that the minimum corresponds to hyperplanes orthogonal to vectors ξ = (ξ¹,...,ξⁿ) ∈ ℂⁿ with |ξ¹| = ... = |ξⁿ|, and the maximum corresponds to hyperplanes orthogonal to vectors with only one non-zero coordinate.

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