Separation of hypersurfaces
Let \(M\) be an embedded compact hypersurface in \(\mathbb{R}^{n+1}\), by \(G_M\) one denotes the Gauss map of \(M\), \(G_M: M\to S^n\). For \(M\), \(N\), two embedded, compact hypersurfaces in \(\mathbb{R}^{n+1}\) the author shows that if \(M^{[n]}=N^{[n]}\), then \(M= N\), where \[ M^{[n]}= \{p\in M:\text{rank }DG_M(p)= n\},\text{ resp. } N^{[n]}=\{p\in N:\text{rank }DG_N(p)=n\}. \] Moreover, he introduces the notion of affine envelope as a link between a hypersurface and its apparent contours and proves that the mapping from all embedded hypersurfaces in \(\mathbb{R}^{n+1}\) into the unions \(\bigcup\{C^M_w: w\in W\}\), (where \(C^M_x\) is the apparent contour of \(M\) in direction \(x\)), under the certain condition on \(W\subset S^n\) is injective. This paper is a generalization of results contained in [\textit{F. Pointet}, C. R. Acad. Sci., Paris, Sér. I 318, No. 12, 1129-1131 (1994; Zbl 0817.57026)].
scientific article; zbMATH DE number 1053894