Convexity of polyhedra

Convexity recognition of the union of polyhedra

Bisubmodular polyhedra, simplicial divisions, and discrete convexity

We consider a class of integer-valued discrete convex functions, called BS-convex functions, defined on integer lattices whose affinity domains are sets of integral points of integral bisubmodular polyhedra. We examine discrete structures of BSconvex functions and give a characterization of BS-convex functions in terms of their convex conjugate functions by means of (discordant) Freudenthal sim...

Beyond convexity: scanning 'non-convex polyhedra'

The enumeration of points contained in an algebraically specified domain is one of the key algorithmic problems in the transformation of scientific programs. However, basic scanning algorithms accept only single convex polyhedra, requiring specialized techniques and causing run-time overhead if the set of points to enumerate is not convex. In this paper, we review the existing approaches to the...

Near convexity, metric convexity, and convexity

It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. 1 Classical and constructive mathematics The arguments in this paper conform to constructive mathematics in the sense of Errett Bishop. This mea...

Graphs of polyhedra; polyhedra as graphs

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedr...