A kinematic formula for affine diameters and affine medians of a convex set (Q1781314)

The author considers the area measure \(S(K_i)\) resp. \(S(K_i')\) of all points of a compact convex set \(K\) in the plane \(R^2\) with \(C^2_+\)-boundary which lie on precisely \(i\) chords of \(K\) with parallel tangent lines at the ends (affine diameters) resp. \(i\) chords of \(K\) halving the area of \(K\) (affine medians). By topological (Sard's lemma) and integral geometric methods he proves 1) \(S(K_2)= S(K_4)=\cdots= S(K_\infty)= 0\) and 2) \({S(\Delta K)\over 4}\leq S(K_1)+ 3S(K_3)+ 5S(K_5)+\cdots\leq {S(\Delta K)\over 2}\) resp. \(1')\) \(S(K_2')= S(K_4')=\cdots= S(K_\infty')= 0\) and \(2')\) \(S(K_1')+ 3S(K_3')+ 5S(K_5')+\cdots\leq {S(\Delta K)\over 4}\). Hereby \(S(\Delta K)\) denotes the area of the difference body \(\Delta K\) of \(K\). Some corollaries follow. Remark: 1) also holds in \(n\) dimensions.