How Brunn-Minkowski inequality is used in Machine Learning part2

Abstract : The present paper investigates the sub-Riemannian version of the equivalence between the curvature-dimension conditions and strong Brunn-Minkowski inequalities in the sub-Riemannian Heisenberg group Hn. We adopt the optimal transport and approximation of Hn developed by Ambrosio and Rigot [1] and combine the celebrated works by M. Magnabosco, L. Portinale and T. Rossi [17] to confirm this. 2.A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories (arXiv) Author : <a href="https://arxiv.org/search/?searchtype=author&query=Jin%2C+R" rel="noopener ugc nofollow" target="_blank">Rongrong Jin</a>, <a href="https://arxiv.org/search/?searchtype=author&query=Lu%2C+G" rel="noopener ugc nofollow" target="_blank">Guangcun Lu</a> Abstract : In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012 <a href="https://medium.com/@monocosmo77/how-brunn-minkowski-inequality-is-used-in-machine-learning-part2-d364b5f673f1">Click Here</a>