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

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
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