Does polynomial parallel volume imply convexity?
For a set \(A \subset \mathbb R^d\) let \(A_r\) denote the set of points with distance at most \(r\) from \(A\). The Steiner formula says that the volume of \(A_r\), \(V(A_r)\), is a polynomial in \(r\) if \(A\) is convex. This paper is about the reverse question. It is shown that if \(d=2\) and if \(V(A_r)\) is polynomial in \(r\), then \(A\) is convex. This is not true for \(d\geq 3\).
scientific article; zbMATH DE number 2078026