Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension

By Emanuel Milman. 2014

68 citations recorded by the source corpus.

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \in [n,\infty]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound, and discover a new case yielding a \emph{single} model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty]$): a (positively curved) sphere of (possibly negative) dimension $N \in (-\infty,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $N$-dimensional Cheeger isoperimetric inequality, and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincar\'e inequality uniformly for all $N \in (-\infty,1-\epsilon]$, and enjoy a two-level concentration of the type $\exp(-\min(t,t^2))$. Our main technical tool is a generalized version of the Heintze--Karcher theorem, which we extend to the range $N \in (-\infty,1)$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/97bdc226d8e0b654bed2c469d9dc598edb30b895

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In Applied Mathematics

Citations in the corpus

A warped product version of the Cheeger-Gromoll splitting theorem (2015, methodology): “on inequalities on Euclidean space with N <0 was investigated in the 1970s by Borell [Bor74] and Brascamp-Lieb[BL76]. We should also caution the reader that our deﬁnition of CD(λ,N), which matches [Mila], is not equivalent to the deﬁnition given by Bakry-Emery [BE85] in the range´ N∈ [0,n), see [”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “In the case of Riemannian and weighted Riemannian manifolds satisfying a strictly positive lower bound on the Ricci curvature, sharp Lévy-Gromov-type isoperimetric inequalities have been obtained byMilman [45, 46].”
The globalization theorem for the Curvature-Dimension condition (2016, background): “dition serves as a generalization of the classical condition in the non-weighted Riemannian setting of having Ricci curvature bounded below by K ∈ R and dimension bounded above by N ∈ [1,∞] (see e.g. [55, 59] for further possible extensions). Numerous consequences of this condition have been obtaine”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, background): “We expect this gap in the literature to be quickly ﬁlled (in fact, concurrently to posting our work on the arXiv, Ohta [39] has posted a ﬁrst attempt of a systematic treatise of the range N ≤ 0, and subsequently other authors have also begun treating this extended range [32, 15, 50, 14, 33]).”
Needle decompositions and isoperimetric inequalities in Finsler geometry (2015, background): “oy the same isoperimetric inequality I(M,g,m)(θ) ≥IK,N,D(θ) for all θ∈[0,1] (1.2) regardless the dimension nof the spaces (we remark that, to be precise, the case of N∈(0,1) and D<∞was excluded in [Mi2], see [Mi2, Remark 1.5]). Furthermore, (1.2) is sharp in all parameters K, N and Din all dimens”