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

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...

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...

All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called “intersection operator”—an operator that played a critical role in the solution of the Busemann-Petty problem. Introduction. A function Z defined on the set K of convex bodies (that is, of conve...

We study succinct approximation of functions that have noisy oracle access. Namely, construction of a succinct representation of a function, given oracle access to an L-approximation of the function, rather than to the function itself. Specifically, we consider the question of the succinct representation of an approximation of a convex function φ that cannot be accessed directly, but only via o...

This paper presents the concepts of (L,M)-remotehood spaces and (L,M)-convergence in framework (L,M)-fuzzy convex spaces. Firstly, it is shown that category isomorphic to Secondly, proved can be embedded as a reflective subcategory. Finally, preconvex (L,M)- convergence are introduced

Finding the minimum vertex distance between two dis-joint convex polygons in linear time. [6] Supowit, K. J.: The relative neighborhood graph, with an application to minimum spanning trees. A note on the all-nearest-neighbor problem for convex polygons. [10] O'Rourke, J., et al.: A new linear algorithm for intersecting convex polygons.Since P nw and Q R are two linearly separable convex polygon...

The cross covariogram gK,L of two convex sets K, L ⊂ R n is the function which associates to each x ∈ R the volume of the intersection of K with L + x. The problem of determining the sets from their covariogram is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse probl...