×

The disintegration of the Lebesgue measure on the faces of a convex function. (English) Zbl 1202.28003

A convex function from \(\mathbb{R}^n\) to \(\mathbb R\) has a natural decomposition: the decomposition into the interiors of its faces. The authors prove a disintegration theorem for Lebesgue measure restricted to the graph of such a function and show in particular that the measure on a \(k\)-dimensional face is equivalent to the \(k\)-dimensional Hausdorff dimension.
Reviewer: K. P. Hart (Delft)

MSC:

28A50 Integration and disintegration of measures
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alberti, G.; Ambrosio, L., A geometrical approach to monotone functions in \(R^n\), Math. Z., 230, 2, 259-316 (1999) · Zbl 0934.49025
[2] Alberti, G.; Ambrosio, L.; Cannarsa, P., On the singularities of convex functions, Manuscripta Math., 76, 3-4, 421-435 (1992) · Zbl 0784.49011
[3] G. Alberti, B. Kirchheim, D. Preiss, personal communication in [4]; G. Alberti, B. Kirchheim, D. Preiss, personal communication in [4]
[4] Ambrosio, L.; Kirchheim, B.; Pratelli, A., Existence of optimal transport maps for crystalline norms, Duke Math. J., 125, 2, 207-241 (2004) · Zbl 1076.49022
[5] Ambrosio, L.; Pratelli, A., Existence and stability results in the \(L^1\) theory of optimal transportation, (Optimal Transportation and Applications. Optimal Transportation and Applications, Martina Franca, 2001. Optimal Transportation and Applications. Optimal Transportation and Applications, Martina Franca, 2001, Lecture Notes in Math., vol. 1813 (2003), Springer-Verlag: Springer-Verlag Berlin), 123-160 · Zbl 1065.49026
[6] Bianchini, S.; Caravenna, L., On the extremality, uniqueness and optimality of transference plans, Bull. Inst. Math. Acad. Sin. (N.S.), 4, 4, 353-454 (2009) · Zbl 1207.90015
[7] S. Bianchini, M. Gloyer, On the Euler Lagrange equation for a variational problem: the general case II, Math. Z., doi:10.1007/s00209-009-0547-2, in press; S. Bianchini, M. Gloyer, On the Euler Lagrange equation for a variational problem: the general case II, Math. Z., doi:10.1007/s00209-009-0547-2, in press · Zbl 1221.49035
[8] L. Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Z., doi:10.1007/s00209-010-0677-6, in press; L. Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Z., doi:10.1007/s00209-010-0677-6, in press · Zbl 1229.49050
[9] Ewald, G.; Larman, D. G.; Rogers, C. A., The directions of the line segments and of the \(r\)-dimensional balls on the boundary of a convex body in Euclidean space, Mathematika, 17, 1-20 (1970) · Zbl 0199.57002
[10] Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag Berlin · Zbl 0176.00801
[11] Feldman, M.; McCann, R., Monge’s transport problem on a Riemannian manifold, Trans. Amer. Math. Soc., 354, 1667-1697 (2002) · Zbl 1038.49041
[12] Hoffmann-Jørgensen, J., Existence of conditional probabilities, Math. Scand., 28, 257-264 (1971) · Zbl 0247.60001
[13] Klee, V.; Martin, M., Semicontinuity of the face-function of a convex set, Comment. Math. Helv., 46, 1-12 (1971) · Zbl 0208.14901
[14] Larman, D. G., A compact set of disjoint line segments in \(R^3\) whose end set has positive measure, Mathematika, 18, 112-125 (1971) · Zbl 0223.28017
[15] Larman, D. G., On a conjecture of Klee and Martin for convex bodies, Proc. Lond. Math. Soc. (3), 23, 668-682 (1971) · Zbl 0245.52003
[16] Larman, D. G.; Rogers, C. A., Increasing paths on the one-skeleton of a convex body and the directions of line segments on the boundary of a convex body, Proc. Lond. Math. Soc. (3), 23, 683-698 (1971) · Zbl 0234.52003
[17] Morgan, F., Geometric Measure Theory: A Beginner’s Guide (2000), Academic Press Inc. · Zbl 0974.49025
[18] Pachl, J. K., Disintegration and compact measures, Math. Scand., 43, 1, 157-168 (1978/1979) · Zbl 0402.28006
[19] Pavlica, D.; Zajíček, L., On the directions of segments and \(r\)-dimensional balls on a convex surface, J. Convex Anal., 14, 1, 149-167 (2007) · Zbl 1127.52005
[20] Rockafellar, R. T., Convex Analysis, Princeton Math. Ser., vol. 28 (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0229.90020
[21] Sudakov, V. N., Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math., 2, 1-178 (1979), Number in Russian series statements 141 (1976)
[22] Trudinger, N. S.; Wang, X. J., On the Monge mass transfer problem, Calc. Var. Partial Differential Equations, 13, 19-31 (2001) · Zbl 1010.49030
[23] Zajíček, L., On the points of multiplicity of monotone operators, Comment. Math. Univ. Carolin., 19, 1, 179-189 (1978) · Zbl 0404.47025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.