An Alexandrov type theorem for \(k\)-convex functions. (English) Zbl 1082.26010

Chaudhuri, Nirmalendu; Trudinger, Neil S.

Bull. Aust. Math. Soc. 71, No. 2, 305-314 (2005).

A continuously twice differentiable function \(u\) on an open subset \(\Omega\) of \(\mathbb R^{n}\) is called \(k\)-convex (\(k=1,\dots,n\)) if for each \(j=1,2,\dots,k\) the sum of the principal minors of order \(j\) of the Hessian matrix of \(u\) is nonnegative on \(\Omega\). If \(u\) is just upper semi-continuous, it is called \(k\)-convex if every quadratic polynomial \(q\) for which the difference \(u-q\) has a finite local maximum in \(\Omega\) is \(k\)-convex in the above sense. The main result in the paper states that, for \(n\geq2\) and \(k>\frac{n}{2}\), every \(k\)-convex function is twice differentiable almost everywhere.

Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)

MSC:

26B25	Convexity of real functions of several variables, generalizations
26B05	Continuity and differentiation questions
49J52	Nonsmooth analysis
28A15	Abstract differentiation theory, differentiation of set functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46F10	Operations with distributions and generalized functions

Keywords:

Alexandrov theorem; generalized convexity; twice differentiability

References:

[1]	Giusti, Minimal surfaces and functions of bounded variation (1984) · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0
[2]	Hörmander, Notions of Convexity (1994)
[3]	DOI: 10.1007/BF02392544 · Zbl 0654.35031 · doi:10.1007/BF02392544
[4]	Alexandrov, Leningrad State University Annals [Uchenye Zapiski] Math. Ser. 6 pp 3– (1939)
[5]	Ziemer, Weakly differentiable functions (1989) · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
[6]	DOI: 10.1090/S0894-0347-05-00475-3 · Zbl 1229.53049 · doi:10.1090/S0894-0347-05-00475-3
[7]	DOI: 10.1353/ajm.2002.0012 · Zbl 1067.35023 · doi:10.1353/ajm.2002.0012
[8]	DOI: 10.1006/jfan.2001.3925 · Zbl 1119.35325 · doi:10.1006/jfan.2001.3925
[9]	DOI: 10.2307/121089 · Zbl 0947.35055 · doi:10.2307/121089
[10]	Trudinger, Topol. Methods Nonlinear Anal. 10 pp 225– (1997)
[11]	Rudin, Real and complex analysis (1987)
[12]	DOI: 10.1080/03605309708821299 · Zbl 0883.35035 · doi:10.1080/03605309708821299
[13]	Krylov, Nonlinear elliptic and parabolic equations of second order (1987) · Zbl 0619.35004 · doi:10.1007/978-94-010-9557-0
[14]	Klimek, Pluripotential theory (1991)
[15]	Ivochkina, Math Sb. (N.S.) 128 pp 403– (1985)
[16]	Evans, Measure theory and fine properties of functions (1992) · Zbl 0804.28001