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Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki (2011)

Colloquium Mathematicae

We show that the cardinality of a compact convex set W in a topological linear space X satisfies the condition that = . We also establish some relations between the cardinality of W and that of extrW provided X is locally convex. Moreover, we deal with the cardinality of the convex set E(μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of extrE(μ).

Cauchy problems for discrete affine minimal surfaces

Marcos Craizer, Thomas Lewiner, Ralph Teixeira (2012)

Archivum Mathematicum

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...

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