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Structural properties of singularities of semiconcave functions. (English) Zbl 0957.26002
Let \(A\subseteq \mathbb{R}^n\). A function \(u:A\to \mathbb{R}\) is called semiconcave if, for some constant \(C\), \(u-C|\cdot|^2\) is concave on every convex subset of \(A\) (\(|\cdot|\) denoting the Euclidean norm). If \(\Omega\subseteq \mathbb{R}^n\) is open, one says that \(u:\Omega\to \mathbb{R}\) is locally semiconcave when it is semiconcave on every compact \(A\subseteq \Omega\). The authors describe the structure of the set of singular (i.e., non-differentiability) points of a locally semiconcave function. They give conditions under which a singular point belongs to a Lipschitz arc (more generally, to a Lipschitz set of given dimension \(\nu\geq 1\)) that consists of singular points. They also discuss the special case of distance functions and, in particular, give a characterization of their isolated singular points.

MSC:
26B25 Convexity of real functions of several variables, generalizations
26B05 Continuity and differentiation questions
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