On directional convexity
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Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: Rd → R is called D-convex, where D is a set of vectors in Rd, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ Rd, denoted by coD(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R2 where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from coD(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R3 (where D is the orthonormal basis of R3), we construct arbitrarily large finite sets A with coD(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices.
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