Let \(\textrm{CB}_0(\mathbb R^n)\) be the set of convex bodies in \(\mathbb R^n\) containing the origin as an interior point. Let \(K^\circ \) denote the polar convex body of a given \(K\in \textrm{CB}_0(\mathbb R^n)\) . Convex bodies of constant width form a rich theory that goes back at least to the time of Euler and continues to attract attention in the convex geometry community. The polarity operation \(K\mapsto K^\circ \) is an involution on \(\textrm{CB}_0(\mathbb R^n)\) that usually destroys the property of being of constant width. In the planar case, think for instance of a Reuleaux triangle whose center is placed at the origin. The polar convex body of such a Reuleaux triangle is not of constant width. It is then natural to ask which is exactly the property of K that ensures width constancy of \(K^\circ \) . This work gravitates around this issue. We introduce and study the concept of the cowidth function of a convex body. Among other results, we prove that \(K^\circ \) is of constant width if and only if K is of constant cowidth. This result is extended to a higher degree of generality by introducing the direct and inverse power-width functions of a convex body.
