The theory of inequalities for convex bodies related to the isoperimetric inequalities has been revived considerably in the last decade. The author contributes several new and interesting results. We mention two of them.
Let A be a convex body in \({\mathbb{R}}^ n\) having a positive continuous curvature function, let \(\pi\) K be the projection body of a convex body K, and let V(K) denote the volume, \(V_ 1(A,B):=V(A,...,A,B)\) a mixed volume.
Theorem: \(V(K)^{n-1}\Omega (A)^{n+1}\leq n(n\omega_ n/\omega_{n- 1})^ n V^ n_ 1(A,\pi K),\) the equality sign holding if and only if A and the polar body \(K^*\) of K are homothetic ellipoids.
Let c be the centroid of K. Then the centroid body \(\Gamma\) K of K is defined by the supporting function \(V(K)^{-1}\int_{-c+K}| x\cdot y| dy.\) Theorem: If K, \(\bar K\) are convex bodies in \({\mathbb{R}}^ n\), then \(V(\pi \bar K)V(K)\leq ((n+1)/2)^ n \omega^ 2_ n V^ n_ 1(\bar K,\Gamma K)\) with equality if and only if K and the polar body of \(\pi\) K are homothetic.