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 subset of or equal to R-d, the fu nctional 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-conve x hulls, e.g., a Krein-Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for compu ting 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 e quations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construc t a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answeri ng a question of K. Zhang on the existence of higher-dimensional nontr ivial configurations of points and matrices).