You can recognize the shape of a figure from its shadows! (Q1906237)

The (divergent beam) shadow picture of a plane convex body \({\mathcal F}\) at the point \(P\) is the measure of the visual angle of \({\mathcal F}\) at \(P\). Let \(C_1\) and \(C_2\) be closed convex domains in the Euclidean plane with \(C^2\) boundaries \(\partial C_1\) and \(\partial C_2\) intersecting each other in nonzero angles. For the special case of two strictly convex plane bodies \({\mathcal F}_1\) and \({\mathcal F}_2\) with \(C^2\) boundaries being in the interior of \(C_1 \cap C_2\), the author proves the following theorem: If \({\mathcal F}_1\) and \({\mathcal F}_2\) have the same shadow picture at each point of \(\partial C_1 \cup \partial C_2\), then they coincide. The proof uses methods of \textit{K. J. Falconer} [Proc. Lond. Math. Soc., III. Ser. 46, 241-262 (1983; Zbl 0502.52005)] and \textit{C. A. Rogers} [Geom. Dedicata 10, 73-78 (1981; Zbl 0452.52001)].