Discrete & Computational Geometry

, Volume 25, Issue 3, pp 389–403

On directional convexity

  • J. Matoušek

DOI: 10.1007/s004540010069

Cite this article as:
Matoušek, J. Discrete Comput Geom (2001) 25: 389. doi:10.1007/s004540010069


Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: RdR is called D-convex, where D is a set of vectors in Rd, if its restriction to each line parallel to a nonzero vD is convex. The D-convex hull of a compact set ARd, 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 AR2 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.

Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • J. Matoušek
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic