Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Steiner type formulae and weighted measures of singularities for semi-convex functions

Authors: Andrea Colesanti and Daniel Hug
Journal: Trans. Amer. Math. Soc. 352 (2000), 3239-3263
MSC (2000): Primary 26B25, 52A41; Secondary 28A78, 52A20, 49J52, 49Q15
Published electronically: March 21, 2000
MathSciNet review: 1751449
Full-text PDF

Abstract | References | Similar Articles | Additional Information


For a given convex (semi-convex) function $u$, defined on a nonempty open convex set $\Omega\subset\mathbf{R}^n$, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for $r\in \{0,\ldots,n\}$, the $r$-th coefficient measure of the local Steiner formula for $u$, restricted to the set of $r$-singular points of $u$, is absolutely continuous with respect to the $r$-dimensional Hausdorff measure, and that its density is the $(n-r)$-dimensional Hausdorff measure of the subgradient of $u$.

As an application, under the assumptions that $u$ is convex and Lipschitz, and $\Omega$ is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of $r$-singular points of $u$. Such estimates depend on the Lipschitz constant of $u$ and on the quermassintegrals of the topological closure of $\Omega$.

References [Enhancements On Off] (What's this?)

  • 1. G. Alberti, L. Ambrosio, and P. Cannarsa,
    On the singularities of convex functions,
    Manuscr. Math. 76 (1992), 421-435. MR 94c:26017
  • 2. R. B. Ash,
    Measure, Integration, and Functional Analysis,
    Academic Press, New York, 1972. MR 55:8281
  • 3. V. Bangert,
    Sets with positive reach,
    Arch. Math. 38 (1982), 54-57. MR 83k:53058
  • 4. F. H. Clarke,
    Optimization and Nonsmooth Analysis,
    Canadian Mathematical Society, Wiley-Interscience Publication, New York, 1983. MR 85m:49002
  • 5. D. L. Cohn,
    Measure Theory,
    Birkhäuser Boston, Boston, 1980. MR 81k:28001
  • 6. A. Colesanti,
    A Steiner type formula for convex functions,
    Mathematika 44 (1997), 195-214. MR 98h:52018
  • 7. A. Colesanti and C. Pucci,
    Qualitative and quantitative results for sets of singular points of convex bodies,
    Forum Math. 9 (1997), 103-125. MR 98c:32004
  • 8. H. Federer,
    Curvature measures,
    Trans. Am. Math. Soc. 93 (1959), 418-491. MR 22:961
  • 9. H. Federer,
    Geometric Measure Theory,
    Springer, Berlin, 1969. MR 41:1976
  • 10. J. H. G. Fu,
    Tubular neighborhoods in Euclidean spaces,
    Duke Math. J. 52 (1985), 1025-1046. MR 87f:57019
  • 11. D. Hug,
    Generalized curvature measures and singularities of sets with positive reach, Forum Math. 10 (1998), 699-728. MR 99j:52004
  • 12. P. Kohlmann,
    Curvature measures and Steiner formulae in space forms,
    Geom. Dedicata 40 (1991), 191-211. MR 93d:53070
  • 13. R. T. Rockafellar,
    Generalized subgradients in mathematical programming,
    in: Mathematical Programming - The State of the Art, A. Bachem, M. Grötschel and B. Korte (eds.), Proc. of the 11th International Symp. on Mathematical Programming, Bonn, 1982, Springer, Berlin, 1983, pp. 368-390. MR 85b:90089
  • 14. R. Schneider,
    Convex Bodies: The Brunn-Minkowski Theory,
    Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1993. MR 94d:52007
  • 15. N. S. Trudinger,
    Isoperimetric inequalities for quermassintegrals,
    Ann. Inst. H. Poincaré, Analyse non Linéaire 11 (1994), 411-425. MR 95k:52013
  • 16. M. Zähle,
    Integral and current representation of Federer's curvature measures,
    Arch. Math. 46 (1986), 557-567. MR 88a:53072

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26B25, 52A41, 28A78, 52A20, 49J52, 49Q15

Retrieve articles in all journals with MSC (2000): 26B25, 52A41, 28A78, 52A20, 49J52, 49Q15

Additional Information

Andrea Colesanti
Affiliation: Universitá Degli Studi di Firenze, Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/A, 50134 Firenze, Italy

Daniel Hug
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstraße 1, D-79104 Freiburg i. Br., Germany

Keywords: Steiner formula, convex function, semi-convex function, singularities, weighted Hausdorff measures, subgradient map, unit normal bundle, non-smooth analysis
Received by editor(s): December 30, 1996
Published electronically: March 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society