A width-diameter inequality for convex bodies (Q1056387)

If \(V\), \(V^*\) are the volumes of a centrally convex body and of its polar body, then according to Blaschke and Santaló \(VV^*\le \omega_n^2\), where \(\omega_n\) is the volume of the unit ball in \(R^n\). Applying this inequality to the vector domain of an arbitrary convex body the author obtains \(M_n[D_k] \le M_{-n}[b_k]\) where \(D_k(u)\), \(b_k(u)\) are diameter and width in the direction \(u\), respectively, and \(M_n[f] = [(n\omega_n)^{-1} \int f^n(u)\,dS(u)]^{1/n}\), the integral extended over the unit sphere. This inequality leads to strengthened versions of several known inequalities (of Herda, Chakerian, Lutwak, Sangwine-Yager) and is related to an inequality of Goodey.