Reduced convex bodies in Euclidean space—A survey

A convex body RR in Euclidean space EdEd is called reduced if the minimal width Δ(K)Δ(K) of each convex body K⊂RK⊂R different from RR is smaller than Δ(R)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 E2E2 are regular odd-gons. We know a relatively large amount on reduced convex bodies in E2E2. Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3d≥3 we do not even have tools which permit us to recognize what the boundary of RR looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in EdEd, recall some striking related research problems, and put a few new questions.