On Functional Separately Convex Hulls

Abstract

Let D be a set of vectors in R d . A function f: R d → R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆ R d , the functional D-convex hull of A, denoted by co D (A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing co D (A) for a finite point set A (in any fixed dimension) in the case of D being a basis of R d (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices).