How Brunn-Minkowski inequality is used in Machine Learning part3

<ol> <li>Brunn-Minkowski inequality for θ-convolution bodies via Ball’s bodies(arXiv)</li> </ol> Author : Brunn-Minkowski inequality for θ-convolution bodies via Ball’s bodies Abstract : We consider the problem of finding the best function φn:[0,1]→R such that for any pair of convex bodies K,L∈Rn the following Brunn-Minkowski type inequality holds |K+θL|1n≥φn(θ)(|K|1n+|L|1n), where K+θL is the θ-convolution body of K and L. We prove a sharp inclusion of the family of Ball’s bodies of an α-concave function in its super-level sets in order to provide the best possible function in the range (34)n≤θ≤1, characterizing the equality cases 2.Brunn-Minkowski inequalities for path spaces on Riemannian surfaces (arXiv) Author : <a href="https://arxiv.org/search/?searchtype=author&query=Assouline%2C+R" rel="noopener ugc nofollow" target="_blank">Rotem Assouline</a> Abstract : We define Minkowski summation with respect to a path space on a manifold, extending the well-known notion of geodesic Minkowski sum. For path spaces on two-dimensional Riemannian manifolds consisting of constant-speed curves, we give necessary and sufficient conditions under which this Minkowski summation satisfies a local Brunn-Minkowski inequality. <a href="https://medium.com/@monocosmo77/how-brunn-minkowski-inequality-is-used-in-machine-learning-part3-91dcb61e9041">Visit Now</a>