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Regularity of shadows and the geometry of the singular set associated to a Monge-Ampère equation. (English) Zbl 1365.53005

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region or a shadow, for short. The shadows were studied by many mathematicians working in differential geometry, convex geometry, geometric combinatorics, and functional analysis. The authors present the contribution of their predecessors in detail and prove the following sharp regularity results for the boundary of the shadow in every direction of illumination:
(1) For a strictly convex domain in \(\mathbb R^n\), the boundary of the shadow generated by parallel illumination is locally a continuous graph in every direction.
(2) There exist a convex set and a direction so that the shadow boundary generated by parallel illumination is not locally a graph.
(3) For a \(p\)-uniformly convex \(C^{1,\alpha}\) domain in \(\mathbb R^n\), with \(\alpha\in (0, 1]\) and \(p\geq 2\), the boundary of the shadow generated by parallel illumination is locally a \(C^{0,\frac{\alpha}{p-1}}\) graph in every direction.
(4) For every \(\alpha\in (0, 1]\) and \(p\geq 2\), there exist a \(C^{\infty}\)-smooth convex set and a direction so that the shadow boundary generated by parallel illumination in that direction is in \(C^{0,\beta}\setminus C^{0,\frac{\alpha}{p-1}}\) for some \(\beta<\frac{\alpha}{p-1}\).
(5) For a 2-uniformly convex \(C^{k+1}\) domain in \(\mathbb R^n\), \(k\geq 1\), the boundary of the shadow generated by parallel illumination is locally a \(C^k\) graph.
The authors also address the problem of regularity of the region generated by orthogonally projecting a convex domain onto another. The precise statement of the problem is as follows: Given two convex domains \(\Omega\subset\mathbb R^n\) and \(\Lambda\subset\mathbb R^n\), if \(P_{\Lambda}(\Omega)\) denotes the orthogonal projection of \(\Omega\) onto \(\Lambda\), then how smooth is \(\partial(P_{\Lambda}(\Omega)\cap\partial\Lambda)\)? This problem was previously studied by different mathematicians. After discussing their results, the authors present their own ones that read as follows:
(i) Let \(\Omega\subset\mathbb R^n\) be a bounded strictly convex domain and \(\Lambda\subset\mathbb R^n\) be a convex domain whose boundary is \(C^{1,1}\). If the closures of the sets \(\Omega\) and \(\Lambda\) are disjoint, then \(\partial(P_{\Lambda}(\Omega))\) is finitely \((n-2)\)-rectifiable.
(ii) The disjointness assumption in (i) is necessary: there exist two bounded convex domains \(\Omega\) and \(\Lambda\) in \(\mathbb R^2\) for which \(\mathcal{H}^0(\partial(P_{\Lambda}(\Omega)\cap\partial\Lambda))=\infty\), where \(\mathcal{H}^0\) stands for the 0-dimensional Hausdorff measure.
(iii) If \(\Omega\) and \(\Lambda\) are \(C^{k+1}\) convex domains in \(\mathbb R^n\) with disjoint closures, \(k\geq 1\), and \(\Omega\) is bounded and 2-uniformly convex, then \(\partial P_{\Lambda}(\Omega)\) is an \((n-2)\)-dimensional \(C^k_{\mathrm{loc}}\) graph.
As an application of the above results, the authors study the geometry and Hausdorff dimension of the singular set of the free boundary of functions related to a Monge-Ampère equation which appears in the optimal partial transport problem. More precisely, they connect the shadow boundaries with the singular set and show that an upper estimate for the Hausdorff dimension of the singular set may be obtained under a strict convexity assumption only.
The results obtained and their proofs highlight the interplay between the shadow generated by parallel illumination, the shadow generated by orthogonal projections on a convex set, and the Monge-Ampère free boundary problem arising in optimal transport theory.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A05 Surfaces in Euclidean and related spaces
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52A10 Convex sets in \(2\) dimensions (including convex curves)
49Q20 Variational problems in a geometric measure-theoretic setting
26B05 Continuity and differentiation questions
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