A Poincaré-type inequality on the Euclidean unit sphere (Q2830099)

The \(L_p\) Minkowski-Firey combination \(t_1K+_p t_2L\) of two convex bodies \(K,L\subset\mathbb R^n\) is defined via its support function \(h\), namely NEWLINE\[NEWLINEh(\cdot)^p=t_1h_K(\cdot)^p+t_2h_L(\cdot)^p.NEWLINE\]NEWLINENEWLINENEWLINEThe motivation to study the second variation of volume for this \(L_p\) combination is that for usual Minkowski addition of bodies this leads to a Poincaré type inequality on the smooth boundary of a convex body [\textit{A. Colesanti}, Commun. Contemp. Math. 10, No. 5, 765--772 (2008; Zbl 1157.52002)].NEWLINENEWLINEThe authors study the second variation by straightforward computation. The main application of the obtained formula is the following inequality.NEWLINENEWLINELet \(1\leq p\leq \infty\). Let \(\psi\in C^1(S^{n-1})\) be any function. Then NEWLINE\[NEWLINE\frac{p-n}{n\omega_n}\left(\int_{S^{n-1}}\psi\right)^2+(n-p)\int_{S^{n-1}}\psi^2\leq \int_{S^{n-1}}(\nabla\psi)^2,NEWLINE\]NEWLINE where \(\omega_n\) is the volume of the unit ball in \(\mathbb R^n\). If \(\int\psi=0\), then we obtain a classical Poincaré inequality.