Quermass-integrals of centro-symmetric universal covers (Q1381325)

A set \({\mathcal D}\) in Euclidean \(n\)-space \(E^n\) is a universal cover if every set \(K \subset E^n\) of diameter one (say) can be covered by a congruent copy of \({\mathcal D}\). The author obtains lower bounds for the quermassintegrals of centrally symmetric convex universal covers in terms of quermassintegrals of bodies of constant width one. It is also proved that among the centrally symmetric convex universal covers in the plane the regular hexagon of width one has the smallest area as well as the smallest perimeter.