## On the singularities of convex functions.(English)Zbl 0784.49011

A rectifiability result is provided for the singular sets of convex and semiconvex functions. In fact, for every real convex or semiconvex function $$u$$ on a convex open subset $$\Omega$$ of $${\mathbf R}^ n$$, and every integer $$k$$ such that $$0< k\leq n$$, one may consider the set $$\Sigma^ k$$ of all points $$x\in\Omega$$ such that the subdifferential of $$u$$ in $$x$$ has dimension greater than or equal to $$k$$. Then the set $$\Sigma^ k$$ is countably $$(n-k)$$-rectifiable, and this means that it may be covered by countably many $$(n-k)$$-dimensional submanifolds of class $$C^ 1$$, except for a $${\mathcal H}^{n-k}$$ negligible subset, where $${\mathcal H}^{n-k}$$ denotes the $$(n-k)$$-dimensional Hausdorff measure.
Reviewer: G.Alberti (Pisa)

### MSC:

 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations 28A78 Hausdorff and packing measures
Full Text:

### References:

 [1] B. Alberti and L. Ambrosio, 1992, to appear [2] L. Ambrosio,Su alcune proprietà delle funzioni convesse, to appear in Atti Accad Naz. Lincei, 1992 [3] L. Ambrosio and P. Cannarsa and H. M., Soner,On the propagation of singularities of semi-convex functions, 1992, to appear · Zbl 0874.49041 [4] J.P. Aubin andH. Frankowska,Set-Valued Analysis, Birkhäuser, Boston, 1990 [5] S. Baldo and E. Ossanna [6] P. Cannarsa andH.M. Soner,On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations., Indiana Univ. Math. J., 36 (1987), pp. 501–524 · Zbl 0612.70016 [7] P. Cannarsa and H. Frankowska,Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim., 29 (1991) · Zbl 0744.49011 [8] F.H. Clarke,Optimization and Nonsmooth Analysis, Wiley & Sons, New York, 1983 · Zbl 0582.49001 [9] E. De Giorgi,Nuovi teoremi relativi alle misure (r)-dimensionali in uno spazio, ad r dimensioni, Ricerche Mat.,4 (1955), pp. 95–113 · Zbl 0066.29903 [10] I. Ekeland andR. Temam,Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976 [11] H. Federer,Geometric Measure Theory, Springer Verlag, Berlin, 1969 · Zbl 0176.00801 [12] W.H. Fleming,The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations, 5 (1969), pp. 515–530 · Zbl 0172.13901 [13] J.H.G. Fu,Monge Ampere functions I, Preprint of the Centre of Mathematical Analysis, Australian National University, 1988 [14] Monge Ampere functions II, Preprint of the Centre of Mathematical Analysis, Australian National University, 1988 [15] H. Ishii andP.L. Lions,Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), pp. 26–78 · Zbl 0708.35031 [16] P.L. Lions,Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, 1982 [17] F. Morgan,Geometric Measure Theor j–A beginner’s guide, Academic Press, Boston, 1988 · Zbl 0671.49043 [18] Y.G. Reshetnyak,Generalized derivative and differentiability almost everywhere, Math. USSR Sbornik,4 (1968), pp. 293–302 · Zbl 0176.12001 [19] L. Simon,Lectures on Geometric Measure Theory. Proceedings of the CMA, Australian National University, Camberra, 1983 · Zbl 0546.49019 [20] A.I. Vol’pert andS.I. Hudjaev,Analysis in classes of discontinuous functions and equations of mathematical Physics, Martinus Nijhoff, Dordrecht, 1980
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.