On quotients of \(i\)th affine surface areas (Q2873178)

Direct application of the following Beckenbach-Descher inequality to the \(i\)th mixed surface area \(\Omega_i\) of a Blaschke combination and a continuous curvature function combination of convex bodies proves the main results in this work.NEWLINENEWLINEThe Beckenbach-Dresher inequality [\textit{E. F. Beckenbach} and \textit{R. Bellman}, Inequalities. Berlin-Göttingen-Heidelberg: Springer-Verlag (1961; Zbl 0097.26502)] establishes that for \(p\geq 1\geq r \geq 0\), \(p\neq r\), \(f,g\) nonnegative measurable functions and \(\phi\) a distribution NEWLINE\[NEWLINE \left(\frac{\int{|f+g|^p\,d\phi}}{\int{|f+g|^r\,d\phi}}\right)^{\frac{1}{p-r}}\leq \left(\frac{\int{f^p\, d\phi}}{\int{f^r\,d\phi}}\right)^{\frac{1}{p-r}}+ \left(\frac{\int{g^p\,d\phi}}{\int{g^r\, d\phi}}\right)^{\frac{1}{p-r}}. NEWLINE\]NEWLINE For \(K,L\) convex bodies, let \(f(K,u)\) denote the continuous curvature function of \(K\), \(K\ddot{+}L\) their Blaschke sum and \(K\breve{+}L\) a convex body whose curvature function satisfies \(f(K\breve{+}L,u)^{-\frac{1}{n+1}}=f(K,u)^{-\frac{1}{n+1}}+f(L,u)^{-\frac{1}{n+1}}\). Then, NEWLINE\[NEWLINE \left(\frac{\Omega_{n-p}(K\ddot{+}L)}{\Omega_{n-r}(K\ddot{+}L)}\right)^{\frac{n+1}{p-r}}\leq \left(\frac{\Omega_{n-p}(K)}{\Omega_{n-r}(K)}\right)^{\frac{n+1}{p-r}}+ \left(\frac{\Omega_{n-p}(L)}{\Omega_{n-r}(L)}\right)^{\frac{n+1}{p-r}}. NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \left(\frac{\Omega_{n-p}(K\breve{+}L)}{\Omega_{n-r}(K\breve{+}L)}\right)^{\frac{1}{r-p}}\leq \left(\frac{\Omega_{n-p}(K)}{\Omega_{n-r}(K)}\right)^{\frac{1}{r-p}}+ \frac{\Omega_{n-p}(L)}{\Omega_{n-r}(L)}^{\frac{1}{r-p}}. NEWLINE\]NEWLINE for values of \(p,r\) fitting in the above mentioned inequality.