Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...

In this note, we prove the following result. Let K ⊂ Rd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λK , by a vector x , depends only on the Euclidean norm of x , then K is a Euclidean ball.

Radon’s theorem asserts that any set S of d + 2 points in [Wd has a partition into two subsets S, and S, such that Conv(S,) rl Conv(S,) # 0, where Conv(Si) denotes the convex hull of Si. This central theorem in the theory of convexity has been extended in many directions. One of the most interesting generalizations is the following theorem, proved by Tverberg in 1966, and now considered as a cl...

Let X be a normed linear space. We investigate properties of vector functions F : [a, b] → X of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity K aF is equal to the variation of F ′ + on [a, b). As an application, we give a simple alternative proof of an unpublished result of the fi...

Closure spaces have been previously investigated by Paul Edelman and Robert Jami-son as \convex geometries". Consequently, a number of the results given here duplicate theirs. However, we employ a slightly diierent, but equivalent, deening axiom which gives a new avor to our presentation. The major contribution is the deenition of a partial order on all subsets, not just closed (or convex) subs...

We consider convex functions on infinite dimensional spaces equipped with measures. Our main results give some estimates of the first and second derivatives of a convex function, where second derivatives are considered from two different points of view: as point functions and as measures.