Inequalities between successive minima and intrinsic volumes of a convex body (Q2638531)

The second theorem of Minkowski establishes a relation between the successive minima \(\lambda_ i(K)\) and the volume V(K) of a complex body \(K\subset E^ d\). Based on this theorem we prove for \(i=1,...,d\) the following relations between the successive minima and intrinsic volumes \(V_ i(K):\) \[ \lambda_{i+1}(K)\cdot...\cdot \lambda_ d(K)V(K)< 2^{d-i}V_ i(K). \] None of these inequalities can be improved.