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Uniqueness theorems for closed convex surfaces. (English. Russian original) Zbl 0976.53072

Dokl. Math. 59, No. 3, 454-456 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 366, No. 5, 602-604 (1999).
From the text: We prove a theorem that implies various uniqueness theorems for closed convex surfaces. The theorem is as follows.
Theorem 1. Let \(F_1\) and \(F_2\) be arbitrary closed convex surfaces (we do not impose any regularity conditions). Then, under suitable shift of the surfaces and choice of the coordinate system, the support functions \(H_1\) and \(H_2\) of these surfaces satisfy one of the following conditions:
(a) \(H_1\equiv H_2\);
(b) \(H_1- H_2= c|z|\), where \(c=\text{const}\neq 0\);
(c) the surfaces \(F_1\) and \(F_2\) have a common point and a common normal \(n_0\) at this point, and \(|H_1(n)- H_2(n)|\geq c|n- n_0|^2\), where \(c>0\) and \(|n- n_0|<\varepsilon\); \(\varepsilon\) sufficiently small.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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