Some geometric inequalities by the ABP method (Q6623557)

The author applies the so-called Alexandrov-Bakelman-Pucci (ABP) method to prove several geometric inequalities, which is inspired by \textit{S. Brendle}'s recent work [J. Am. Math. Soc. 34, No. 2, 595--603 (2021; Zbl 1482.53012)] that used the ABP method to establish the optimal isoperimetric inequality for co-dimension \(1\) and \(2\) minimal submanifolds in Euclidean space.\N\NThe first main result of the paper states that the logarithmic Sobolev inequality holds on closed co-dimension \(1\) and \(2\) minimal submanifolds of \(\mathbb{S}^{n+m}\), which follows as a direct consequence of a higher co-dimensional counterpart. The second main result of the paper extends the Sobolev-type inequality for positive symmetric tensors on smooth bounded convex domains by \textit{D. Serre} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 5, 1209--1234 (2018; Zbl 1393.35181)] to hold without the convexity assumption on the domain. The third main result of the paper is the establishment of a geometric inequality related to quermassintegrals for closed \((k+1)\)-convex hypersurfaces in \(\mathbb{R}^{n+1}\) when \(0\leq k\leq n-2\).