If a convex body K in R is contained in a convex body L of elliptic type (a curvature image), then it is known that the affine surface area of K is not larger than the affine surface area of L. We prove that the affine surface areas of K and L can only be equal if K = L. 2010 Mathematics Subject Classification: primary 52A10; secondary 53A15

in this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional euclidean space, which are the einstein and möbius gyrovector spaces. we introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and see its interest...

We let R be an o-minimal expansion of a field, V a convex subring, and (R0, V0) an elementary substructure of (R, V ). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V , and we let LR0 be the language L expanded by constants for all elements of R0. Our main result is that (R, V ) considered as an LR0-structure is model com...

which is still the best known estimate for arbitrary convex bodies. Another class consists of convex bodies with small volume ratio with respect to the ellipsoid of minimal volume. This includes the class of zonoids. K. Ball [BA] solved the problem for the duals of zonoids, i.e. unit balls of subspaces of an L1-space, briefly L1-sections. Theorem 1 (K. Ball) For a convex, symmetric body K ⊂ IR ...

Much of the recent literature on risk measures is concerned with essentially bounded risks in L∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on Lp spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk me...

In this paper, we introduce the concept of convex structure in generalized fuzzy metric spaces and proved common fixed point theorems for a pair self-mappings under sufficient contractive type conditions.

Let (A,LA) be a quantum metric space. Then clearly S(A) with the metric ρLA is a compact balanced convex metric space, i.e. the metric ρLA is convex and balanced. Another important property of S(A) is that the R−valued affine continuous functions Af(S(A)) separate the points of S(A). Conversly, let (X, ρ) be a compact balanced convex metric space on which Af(X) separate the points of X. Then Af...