A Banach space X is called polyhedral if the unit ball of each one of its finite-dimensional (equivalently: two-dimensional [6]) subspaces is a polytope. Polyhedral spaces were studied by various authors; most of the structural results are due to V. Fonf. We refer the reader to the surveys [1], [2] for other definitions of polyhedrality, main properties and bibliography. In this paper we present a short alternative proof of the basic result on the structure of the unit ball of the polyhedral space...

We give a full characterization of the closed one-codimensional subspaces of ${c}_{0}$, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented.

Let $X$ be a non-reflexive real Banach space. Then for each norm $|\xb7|$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|\xb7\left|\right)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

For convex continuous functions $f,g$ defined respectively in neighborhoods of points $x,y$ in a normed linear space, a formula for the distance between $\partial f\left(x\right)$ and $\partial g\left(y\right)$ in terms of $f,g$ (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly...

Let $X$ be a normed linear space. We investigate properties of vector functions $F:[a,b]\to X$ of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity ${K}_{a}^{b}F$ is equal to the variation of ${F}_{+}^{\text{'}}$ on $[a,b)$. As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...

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