Can you recognize the shape of a figure from its shadows? (Q1892777)

If \(C\) is a closed convex curve in the plane, then a compact convex set \(K\) enclosed by \(C\) is said to be distinguishable from \(C\) if the angles which \(K\) subtends at the points of \(C\) determine \(K\) uniquely. As is well known, an ellipse \(K\) is not distinguishable from a concentric circle \(C\) with radius \(\sqrt 2 \text{diam} (K)\). The authors here show that, if \(C\) is analytic, then any polygon is distinguishable from \(C\); in consequence, the set of compact convex sets inside \(C\) which are not distinguishable from \(C\) is of first Baire category in the Hausdorff metric topology. The authors also consider some related problems, and pose various questions.