On directional convexity |
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Authors: | J Matoušek |
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Institution: | (1) Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic |
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Abstract: | Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties
of D-convexity. A function f: R
d → R is called D-convex, where D is a set of vectors in R
d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R
d, 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 ⊂ R
2 where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co
D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R
3 (where D is the orthonormal basis of R
3), we construct arbitrarily large finite sets A with co
D(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.
This research was supported by Charles University Grants No. 158/99 and 159/99. |
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Keywords: | |
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