Let |·| be the standard Euclidean norm on ℝ n and let X=(ℝ ,∥·∥) a normed space. A subspace Y⊂X is strongly α-Euclidean if there constant t such that t|y|≤∥y∥≤αt|y| for every y∈Y, say it α-complemented ∥P Y ∥≤α, where P orthogonal projection from X to ∥ denotes operator of with respect X. We give an example space arbitrarily high dimension 2-Euclidean but contains no 2-dimensional both (1+ϵ)-Eu...

K 〈x, y〉 dy = ‖x‖ By ‖·‖ we denote the standard Euclidean norm. The notion of isotropic position was extensively studied by V. Milman and A. Pajor [M-P]. Note that our definition is consistent with [K-L-S]. The normalization in [M-P] is slightly different. If the information about the body K is uncomplete it is impossible to bring it exactly to the isotropic position. So, the definition of the ...

Known properties of ‘‘canonical connections’’ from database theory and of ‘‘closed sets’’ from statistics implicitly define a hypergraph convexity, here called canonical convexity (cconvexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski–Krein–Milman property. Moreover, we compare c-convexity wi...

Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the dual mixed volume, the fundamental concept in the dual Brunn-Minkowski theory. The characterizations are shown to be best possible in the sense that none of t...

the aim of this work is to describe the qualitative behavior of the solution set of a givensystem of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. in order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. this is done by the extension of ...

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the MilmanPettis theorem that uniformly convex Banach spaces are reflexive. Our aim in this note is to present a fully constructive analysis of the Milman-Pettis theorem [11, 12, 9, 13]: a uniformly convex Banach space is reflexive. First, t...

Existence of fixed point in orthogonal metric spaces has been initiated recently by Eshaghi and et al. [On orthogonal sets and Banach fixed Point theorem, Fixed Point Theory, in press]. In this paper, we introduce the notion of the strongly orthogonal sets and prove a genuine generalization of Banach' fixed point theorem and Walter's theorem. Also, we give an example showing that our main theor...

Corresponding to each origin-symmetric convex (or more general) subset of Euclidean n-space R, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace ofR. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the ...

We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the nonsymmetric setting an inequality of K. Ball [2] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for logconcave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman [1], wi...

Violator Spaces were introduced by J. Matoušek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by antiexchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator spaces ...