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The Brunn–Minkowski inequality implies the CD condition in weighted Riemannian manifolds [PDF]

Nonlinear Analysis, 2022
Mattia Magnabosco, Lorenzo Portinale, Tommaso Rossi +2 more
semanticscholar +1 more source

The Brunn–Minkowski–Firey inequality for nonconvex sets

Advances in Applied Mathematics, 2012
In this short note, the authors first extend the definition of Minkowski-Firey $L_p$-combinations from convex bodies to arbitrary subsets of Euclidean space, and then prove the Brunn-Minkowski-Firey inequality for compact (not necessarily convex) sets of $\mathbb{R}^n$.
Lutwak, Erwin, Yang, Deane, Zhang, Gaoyong +2 more
openaire +1 more source

Sobolev-to-Lipschitz property on QCD -spaces and applications. [PDF]

Math Ann, 2022
Dello Schiavo L, Suzuki K.
europepmc +1 more source

The Brunn–Minkowski inequality for volume differences

Advances in Applied Mathematics, 2004
Suppose that $K$, $L$, $D$, $D'$ are compact domains in $\mathbb{R}^n$ such that $D$ and $D'$ are homothetic and convex and $D\subset K$, $D'\subset L$. It is proved (in a more general form) that for the volume $V$ one has \[ ((V(K+ L)- V(D+ D'))^{1/n}\geq (V(K)- V(D))^{1/n}+ (V(L)- V(D'))^{1/n}.
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Almost-Riemannian manifolds do not satisfy the curvature-dimension condition. [PDF]