An inequality for the volume of inscribed ellipsoids (Q909241)

Let K be a convex body in \({\mathbb{R}}^ n\), and let \(x^*\in int K\) be the center of the ellipsoid of the maximal volume inscribed in the body. An arbitrary hyperplane through \(x^*\) divides K into two convex bodies \(K^+\) and \(K^-\). It is shown that \[ w(K^{\pm})/w(K) \leq \max_{d^ 2}\{1/(d^ 2)\cdot \exp (2-[\sqrt{(4d^ 2+1)}]/d^ 2)\}=0.844..., \] where w(.) is the volume of the inscribed ellipsoid. (Note that there is a misprint in the final inequality.)