New volume ratio properties for convex symmetric bodies in \({\mathbb{R}}^ n\)
Let K be any convex, centrally symmetric, bounded and absorbing set in \({\mathbb{R}}^ n\) and let \(K^*\) be the polar body of K. By a(K) denote the product of the volumes \(V(K)\cdot V(K^*)\). It is known that \(b^ 2_ n/n^{n/2}\leq a(K)\leq b^ 2_ n\), where \(b_ n\) denotes the volume of the (Euclidean) unit ball in \({\mathbb{R}}^ n\). The right hand side inequality is due to Blaschke and Santaló. It is attained precisely for ellipsoids. The left hand side inequality is due to Bambah. The authors improve this essentially by establishing the lower bound \(c^ nb^ 2_ n\), where c is an absolute positive constant (Theorem 1). This already comes very close to a conjecture of Mahler which asserts that \(4^ n/n!\) is the (optimal) lower bound. Given a normed space E with unit ball K, let v(E) denote the volume ratio \((V(K)/V(J))^{1/n}\), where j denotes John's ellipsoid of maximal volume contained in K. For cotype-2 spaces E with cotype-2 constant C(E) the following estimate for the volume ratio is established (Theorem 2): v(E)\(\leq cC(E) (\log C(E))^ 4\). This gives an affirmative answer to a question raised by Pelczynski. Theorem 1 admits interesting applications to the Geometry of Numbers: Let K be any bounded 0-symmetric convex body in \({\mathbb{R}}^ n\) and \(K^*\) its polar body. Then there exists an absolute constant \(c_ 1\) such that either \(c_ 1n^{1/2}K\) or \(c_ 1n^{1/2}K^*\) contains a non-trivial point of the integer lattice \({\mathbb{Z}}^ n\). This result is optimal for ellipsoids. Further consequences concern successive minima, mixed volumes and harmonic analysis.
scientific article; zbMATH DE number 4000872
scientific article