Algebraic properties of sets which do not contain straight lines (Q1603027)

Let \(A\) be a set in an \(n\)-dimensional Euclidean space and \(\Gamma(A)\) be its boundary. Theorem 3 of the paper asserts that if \(n>1\) and the convex hull of \(A\) does not contain straight lines, then each point of \(A\) is the midpoint of a line segment determined by some boundary points of \(A\). Simple examples show that each of the hypotheses is necessary. Subsequently one proves that assuming moreover that \(A\) cannot be represented as the union of two subsets at positive distance to each other, then \(A+A\subseteq \Gamma(A)+\Gamma(A)\). The proof idea is to approximate \(A\) by sets consisting of cubes determined by hyperplanes, parallel to the coordinate ones. The actual proof is rather intricate and requires several preparatory results, some of which might be of independent interest.