Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost ${\rm BCD}$ condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the ${\rm BCD}$ framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.
We study complete
n
-dimensional submanifolds immersed in the Gaussian space
$$\mathbb {G}^{n+p},$$
that is, the Euclidean space
$$\mathbb {R}^{n+p}$$
endowed with its standard metric and the Gaussian weight
$$e^{-|x|^2/4}$$
. More precisely, in codimension
$$p\ge 2,$$
we consider submanif...
W. Barboza, H. D. de Lima, Wallace Gomes· Annali di Matematica Pura ed...· 0 citations
The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to E...
A metric space $Z$ gives rise to three natural classes of infinite-dimensional metric spaces associated to $Z$: $p$-Wasserstein spaces of probability measures on $Z$, nonlinear Lebesgue $L^p$-spaces of $Z$-valued maps, and $p$-Gromov-Wasserstein spaces of $Z$-valued kernels. The latter class, referred to as $Z$-Gromov-...
Martin Bauer, Facundo Mémoli, Tom Needham et al.· 0 citations
We determine the optimal order of the scaling parameter in the metric cotype $2$ inequality for complete Alexandrov spaces of nonnegative curvature. It grows linearly with the dimension $n$ of the discrete torus. This answers Question 17 of Eskenazis, Mendel, and Naor. In their sign-vector normalization on $\bigl(\math...
In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $>3- \frac{1}{n-1}$ canon...
We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in\mathbb{R}$ and $N \in (1,\infty)$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXi...
Luca Rizzi, Zhang Ye· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
