The mean circumscribed polygons
Let \(U\) be a finite set of unit vectors in Euclidean plane such that the origin is in the interior of the convex hull of \(U\). Let \(K\) be a convex body. The author considers the intersection \(C_U(K)\) of the closed supporting halfplanes of \(K\) with outer unit normals in \(U\). The oriented angles of those normals are denoted by \(\beta_0,\dots, \beta_{n-1}\), where \(0\leq \beta_0\leq\dots\leq \beta_{n-1}< 2\pi\). Clearly, \(C_U(K)\) is a polygon circumsribed about \(K\). The symbol \(\vartheta (\alpha)\) denotes the rotation by angle \(\alpha\). The so-called mean convex body \(M_U(K)\) of the circumscribed polygons \(C_{\vartheta (\alpha)U} (K)\) is described by its support function \(h(M_U(K), \beta)= \frac{1}{2\pi} \int_0^{2\pi} h(C_{\vartheta (\alpha)U} (K), \beta) d\alpha\). Let \(\Delta= (\delta_1,\dots, \delta_{n-1})\), where \(\delta_i= \beta_i- \beta_{i-1}\) for \(i=1,\dots, n-1\). The author proves that the mapping \(K\to M_{U(\Delta)} (K)\) is injective if and only if \(\sum_{i=1}^n \frac{\cos \delta_i-\cos (k\delta_i)} {\sin \delta_i}\neq 0\) for every natural \(k\geq 2\).
scientific article; zbMATH DE number 1295486