How Brunn-Minkowski inequality is used in Machine Learning part4

Abstract : In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott — Sturm — Villani, and a newly introduced notion that we call strong Brunn — Minkowski inequality SBM(K,N). This condition is a reinforcement of the generalized Brunn — Minkowski inequality BM(K,N), which is known to hold in CD(K,N) spaces. Our result is a first step towards providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which we have recently proved in the framework of weighted Riemannian manifolds. 2.Horocyclic Brunn-Minkowski inequality (arXiv) Author : <a href="https://arxiv.org/search/?searchtype=author&query=Assouline%2C+R" rel="noopener ugc nofollow" target="_blank">Rotem Assouline</a>, <a href="https://arxiv.org/search/?searchtype=author&query=Klartag%2C+B" rel="noopener ugc nofollow" target="_blank">Bo’az Klartag</a> Abstract : Given two non-empty subsets A and B of the hyperbolic plane H2, we define their horocyclic Minkowski sum with parameter λ=1/2 as the set [A:B]1/2&sube;H2 of all midpoints of horocycle curves connecting a point in A with a point in B. These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter 0<λ<1 is defined analogously. We prove that when A and B are Borel-measurable, <a href="https://medium.com/@monocosmo77/how-brunn-minkowski-inequality-is-used-in-machine-learning-part4-f70ab6c7870a">Visit Now</a>