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Discrete & Computational Geometry

, Volume 19, Issue 1, pp 105–130 | Cite as

On Functional Separately Convex Hulls

  • J. Matoušek
  • P. Plecháč

Abstract.

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⊆R d , the functional 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 -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components.

We give a polynomial-time algorithm for computing 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 equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices).

Keywords

Differential Equation Mathematical Modeling Partial Differential Equation Convex Function Convex Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • J. Matoušek
    • 1
  • P. Plecháč
    • 2
  1. 1.Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic matousek@kam.mff.cuni.czCZ
  2. 2.Department of Mathematics, Heriot—Watt University, Edinburgh EH14 4AS, Scotland petr@ma.hw.ac.uk UK

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