Some inequalities about dual mixed volumes of star bodies
The authors present some inequalities involving dual mixed volumes of star bodies. These inequalities are analogous to the corresponding ones for the quermassintegrals of a convex body. There are interesting new examples on the duality existing in Geometric Tomography between projections and sections. Let \(K\) and \(L\) be an arbitrary pair of star bodies in the \(n\)-dimensional Euclidean space; let \(B\) denote the unit ball; then \[ \frac {\Tilde V_k(K \Tilde + L)}{\Tilde V_{k-1}(K \Tilde + L)} \leq \frac {\Tilde V_k(K)}{\Tilde V_{k-1}(K)} + \frac {\Tilde V_k(L)}{\Tilde V_{k-1}(L)} \] for \(k=1,2\), where \(\Tilde V_i(K)\) is the \(i\)th coefficient in volume of the radial Minkowski linear combination of \(i\) copies of \(K\) and \((n-i)\) copies of \(B\). Let \(K\) be a star body and \(L\) a ball in the \(n\)-dimensional Euclidean space; then the inequality above holds for \(k=1,\dots,n\).
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