Duality of convex bodies (Q1904111)

The authors present a new duality relation in the class of convex bodies which has the advantage that congruent bodies have congruent duals. This notion is defined via polarity with respect to a sphere \(S(K)\), uniquely associated to a convex body \(K\), which has the same centre as the minimal shell of \(K\) and radius equal to the geometric mean of the radii of the shell. The particular choice of \(S(K)\) and the properties of the minimal shell imply that this relation is an involution with euclidean balls as self-duals. Moreover, both the new duality relation and the classical one are studied in the context of category theory.