- Sufficient and necessary conditions for the Lp-Brunn-Minkowski inequality conjecture for p\in[0,1)(arXiv)
Author : Shi-Zhong Du
Abstract : The Lp-Brunn-Minkowski inequality palys a central role in the Brunn-Minkowski theory proposed by Firey [13] Developed by Luttwak in the 60’s [26,27] In the 90s, we generalized the classical Brun-Minkowski inequality by the Lp sum of convex fields. This inequality was established by Firey for p>1 and subsequently deduced by Borozky-Lutwak-Yang-Zhang. [5] for pins\in[0,1). (see also [23,7]) The validity of this conjecture was verified for the planar case. [5]and for higher dimensions as p approaches 1, by Chen-Huang-Li-Liu [7]. (See also local version by Kolesnikov-Milman) [23]) In this brief note, we provide a brief discussion that clarifies the equivalence between the complete conjecture and the lower bound of the third eigenvalue of the Alexandrov problem.
2. Minkowski inequality for Horowitz-Meyers geometry (arXiv)
author : Agil Alley, Peiken Hung
overview : We prove sharp inequalities for toroidal hypersurfaces in 3- and 4-dimensional Horowitz-Myers geometry. It extends previous results on the Minkowski inequality in static spacetime to toroidal surfaces of asymptotic hyperbolic manifolds with flat toroidal conformal infinity.
