# On directional convexity

- First Online:

- Received:
- Revised:

DOI: 10.1007/s004540010069

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

## 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 co^{D}(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.