Reduced convex bodies in Euclidean space—A survey

Abstract

AbstractA convex body R in Euclidean space Ed is called reduced if the minimal width Δ(K) of each convex body K⊂R different from R is smaller than Δ(R). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E2 are regular odd-gons. We know a relatively large amount on reduced convex bodies in E2. Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in Ed, recall some striking related research problems, and put a few new questions