On functional separately convex hulls

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 × 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).