An equivalence of dual Brunn-Minkowski inequality for volume differences (Q2841076)

In [\textit{S. Lv}, Geom. Dedicata 145, 169--180 (2010; Zbl 1202.52008)], a dual Brunn-Minkowski inequality for the volume difference was proved, namely, 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 NEWLINE\[NEWLINE \bigl(V(M\tilde{+}M')-V(K\tilde{+}L)\bigr)^{1/n}\geq\bigl(V(M)-V(K)\bigr)^{1/n}+\bigl(V(M')-V(L)\bigr)^{1/n}. NEWLINE\]NEWLINE Here \(V(K)\) denotes the volume of \(K\) and \(\tilde{+}\) represents the radial sum.NEWLINENEWLINEIn the paper under review, the authors obtain an equivalent form of the above inequality, which is a kind of dual Kneser-Süss inequality for volume differences: NEWLINE\[NEWLINE \bigl(V(M\dot{+}M')-V(K\dot{+}L)\bigr)^{(n-1)/n}\geq \bigl(V(M)-V(K)\bigr)^{(n-1)/n}+\bigl(V(M')-V(L)\bigr)^{(n-1)/n}; NEWLINE\]NEWLINE the operation \(\dot{+}\) is the well-known Blaschke addition of (star) bodies.