Convex bodies forming pairs of constant width (Q797834)

Generalizing the notion of constant width, the author calls a pair (X,Y) of non-empty compact convex sets X and Y in Euclidean n-space a pair of constant width, if \(h_ X(u)+h_ y(-u)\) equals a constant for all unit vectors u, where \(h_ X\) and \(h_ Y\) denote the support function of X and Y, respectively. He shows that (X,Y) is a pair of constant width iff \(\Omega_ r(X)=Y\) and \(\Omega_ r(Y)=X\). Here \(\Omega_ r(X)\) denotes the intersection of all balls of radius r with center in X. He also characterizes \(\Omega^ 2_ r(X)\) and proves \(\Omega^ 3_ r(X)=\Omega_ r(X)\) for non-empty X of circumradius \(\leq r\). Finally, he constructs examples for pairs of constant width.