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How smooth is the shadow of a smooth convex body? (English) Zbl 0601.52002

The author investigates the problem stated in the title of the paper. Let A be a convex compact body in \({\mathbb{R}}^ 3\) and \(P: {\mathbb{R}}^ 3\to {\mathbb{R}}^ 2\) a linear surjection. Then: 1) if \(\partial A\in C^ 1\), then \(\partial (P(A))\in C^ 1\); 2) if \(\partial A\in C^ 2\) with Lipschitz continuous second derivatives, then \(\partial (P(A))\) is twice differentiable; 3) if \(\partial A\) is real-analytic, then \(\partial (P(A))\in C^{2+\epsilon}\) for some \(\epsilon >0\). On the other hand, \(\partial A\in C^{\infty}\) does not imply \(\partial (P(A))\in C^ 2.\)
Note that this paper is also published in J. Lond. Math. Soc. 33, 101-109 (1986).
Reviewer: J.Danes

MSC:

52A15 Convex sets in \(3\) dimensions (including convex surfaces)
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