Discrete & Computational Geometry

, Volume 25, Issue 3, pp 389–403 | Cite as

On directional convexity

  • J. Matoušek
Article

Abstract

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.

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References

  1. [AH]
    R. Aumann and S. Hart. Bi-convexity and bi-martingales. Israel. J. Math., 54(2): 159–180, 1986.CrossRefMATHMathSciNetGoogle Scholar
  2. [BFJK]
    K. Bhattacharya, N. B. Firoozye, R. D. James, and R. V. Kohn. Restrictions on microstructure. Proc. Roy. Soc. Edinburgh Sect. A, 124:843–878, 1994.CrossRefMATHMathSciNetGoogle Scholar
  3. [C]
    E. Casadio-Tarabusi. An algebraic characterization of quasiconvex functions. Ricerche Mat., 42:11–24, 1993.MathSciNetGoogle Scholar
  4. [DKMŠ]
    G. Dolzmann, B. Kirchheim, S. Müller, and V. Šverák. The two-well problem in three dimensions. Calc. Var. Partial Differential Equations, 10(1):21–40, 2000.CrossRefMATHMathSciNetGoogle Scholar
  5. [K]
    B. Kirchheim. On the geometry of rank-one convex hulls. Manuscript, Max Planck Institute for Mathematics in the Sciences, Leipzig, 1999.Google Scholar
  6. [KKB]
    B. Kirchheim, J. Kristensen, and J. Ball. Regularity of quasiconvex envelopes. Preprint 72/1999, Max Planck Institute for Mathematics in the Sciences, Leipzig, 1999.Google Scholar
  7. [L]
    B. Letocha. Directional convexity (in Czech). M.Sc. Thesis, Department of Applied Mathematics, Charles University, Prague, 1999.Google Scholar
  8. [Mo]
    C, B. Morrey. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25–53, 1952.CrossRefMATHMathSciNetGoogle Scholar
  9. [MP]
    J. Matoušek and P. Plecháč. On functional separately convex hulls. Discrete Comput. Geom., 19:105–130, 1998.CrossRefMathSciNetGoogle Scholar
  10. [MŠ1]
    S. Müller and V. Šverák. Convex integration for Lipschitz mappings and counterexamples to regularity. Preprint 26/1999, Max Planck Institute for Mathematics in the Sciences, Leipzig, 1999.Google Scholar
  11. [MŠ2]
    S. Müller and V. Šverák. Convex integration with constraints and applications to phase transitions and partial differential equations. J. European Math. Soc, l(4):393–422, 1999.Google Scholar
  12. [Mü1]
    S. Müller. Variational models for microstructure and phase transitions. In Calculus of Variations and Geometric Evolution Problems (S. Hildebrandt et al., eds.), pages 85–210. Lecture Notes in Mathematics, vol. 1713. Springer-Verlag, Berlin, 1999.CrossRefGoogle Scholar
  13. [Mü2]
    S. Müller. Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Not. 1999, 20:1087–1095, 1999.CrossRefGoogle Scholar
  14. [Sc]
    V. Scheffer. Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities. Dissertation, Princeton University, 1974.Google Scholar
  15. [Šv]
    V. Šverák. New examples of quasiconvex functions. Proc. Roy. Soc. Edinburgh Sect. A, 120:185–189, 1992.CrossRefMATHMathSciNetGoogle Scholar
  16. [T]
    L. Tartar. On separately convex functions. In Microstructure and Phase Transition, The IMA Volumes in Mathematics and Its Applications, vol. 54 (D. Kinderlehrer et al., eds.), pages 191–204. Springer-Verlag, Berlin, 1993.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

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

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