Is a complete, reduced set necessarily of constant width? (Q1737114)

A convex body \(K\) in \(\mathbb{R}^{n}\) is of constant width if the distance between any two parallel supporting hyperplanes to \(K\) is the same. A convex body is diametrically complete if any proper superset of \(K\) has a larger diameter than \(K\). In \(\mathbb{R}^{n}\) with the Euclidean norm a convex body has constant width if and only if \(K\) is diametrically complete. In \(\mathbb{R}^{n}\) with a norm different from the Euclidean norm this does not hold in general. A norm on \(\mathbb{R}^{n}\) is called perfect if this equivalence holds. This is the starting point of the paper. A convex body \(K\) is called reduced if any proper convex subset has a smaller minimal width. The authors show that a complete and reduced simplex is of constant width. This also holds for convex bodies with a smooth extreme point.