Dual Brunn-Minkowski inequality for volume differences
The author proves several dual Brunn-Minkowski type inequalities for the volume difference, dual quermassintegral difference, width-integral difference or dual mixed volume difference of star bodies. For instance, it is shown that if \(K,L,M,M'\) are star bodies in \({\mathbb R}^n\) such that \(M'\) is a dilation of \(M\) and \(K\subset M\), \(L\subset M'\), then \[ \bigl(V(M\overset{\sim}{+}M')-V(K\overset{\sim}{+}L)\bigr)^{1/n}\geq \bigl(V(M)-V(K)\bigr)^{1/n}+\bigl(V(M')-V(L)\bigr)^{1/n}, \] \[ \bigl(\widetilde{V}_1(M,M')-\widetilde{V}_1(K,L)\bigr)^n\geq \bigl(V(M)-V(K)\bigr)^{n-1}\bigl(V(M')-V(L)\bigr). \] Here \(V(K)\) denotes the volume of \(K\), \(\widetilde{V}_i(K,L)=\widetilde{V}(K,\overset{(n-i)}{\dots},K,L,\overset{(i)}{\dots},L)\) is the \(i\)-th dual mixed volume of \(K\) and \(L\) and \(\overset{\sim}{+}\) represents the radial sum. In most of these results, equality holds if and only if \(K\) and \(L\) are homothetic, with the same homothety factor as \(M,M'\).
scientific article; zbMATH DE number 5693222