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Quantitative estimates for the Bakry–Ledoux isoperimetric inequality

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In fact, the model space for RicN ≥ N − 1 is a sphere, and some stability estimates with respect to the diameter were essentially used in [CMM]. In the rigidity case, an isoperimetric minimizer is actually given as a sub-level set of the associated guide function (see Theorem 2.8). The use of the inverse Poincare inequality is inspired by [Ma2], where we studied the rigidity problem, and reveals an interesting connection between the isoperimetric inequality and the spectral gap via the guiding function.

The inverse Poincar´e inequality plays an essential role in integrating 1-dimensional estimates on needles into an estimate of M in the proof of the main theorem (precisely, Proposition7.3 . is a key ingredient). From Ric∞≥K >0 we also have a lower bound for the first non-zero eigenvalueλ1 of−∆m asλ1. This is a generalization of the classical Lichnerowicz inequality to the Ric∞ context and equivalent to the Poincar'e inequality.

We refer to [GKKO] for the generalization of Theorem 2.4 to RCD(K,∞)-spaces and to [Ma1] for the case RicN ≥ K > 0 with N < −1, where we have a distorted splitting product of hyperbolic nature instead of isometric splitting. This is true for isoperimetric minimizers based on regularity theory in geometric measure theory (see [Mi1,§2.2] for example). We owe our argument about quantitative isoperimetric inequalities to Klartag's proof in [Kl] of the isoperimetric inequality (Theorem 2.5) by needle decomposition.

2 Estimates (3.3) and (3.4) on the weighting function could be compared to [CMM, Proposition A.3], which, thanks to the finite dimensionality, is in terms of the deficit in the diameter limit (not directly to the deficitδ in the isoperimetric profile as above).

4 Small deficit implies small symmetric difference

As a consequence of Statement 3.2 together with (3.9), the unique minimizer of ψ is close to that of ψg, namely 0 (note that 0∈I indeed holds when δ is small enough because T → ∞ and S→. Therefore, ∂A occurs exactly once near aθ or −aθ, and all other points of ∂A are far from ±aθ.Since the evidence is common, we will assume that ∂A occurs near aθ (as the right-hand end of the component) in the following.

Regarding the connected component A, whose boundary points are far from ±aθ, we can move it (in I) in the direction opposite to aθ, preserving the total mass and thus the symmetrical difference with (−∞, r−m( θ)] , and decreasing circumference.

5 Reverse Poincar´ e inequality on needles

6 Reverse Poincar´ e inequality on M and applications

Decomposition of deficit

Recall from subsection 2.3 that this needle decomposition can be used to prove the isoperimetric inequality I(M,m) ≥ I(R,γ) on M via those on needles (Xq,mq). By [CMM, Lemma 4.1], one can decompose the isoperimetric deficit of A into those of Aq as follows. We note that what we need to take care of is the measurability of P(Aq) in q ∈ Q, then the inequality itself follows from Fatou's lemma.

Reverse Poincar´ e inequality

Since most needles are long and the measure on them is close to the Gaussian measure γ, (6.2) shows that, on most needles, the conductance function u reaches 0 at a point close to the maximum of the density function (minimum of the weight) function ψ). This observation plays an essential role in integrating the estimates on needles (see Proposition 7.3 and the proofs of Proposition 7.4 and Theorem 7.5), and we emphasize it.

Reverse logarithmic Sobolev inequality

For example, Colding [Co1, Co2] showed that an n-dimensional Riemannian manifold (M, g) satisfying Ricg ≥ n−1 is close to the unit sphere Sn in the Gromov-Hausdorf distance if and only if the volume follows ( M) is close to Sn. We refer to [HM, KaMo] for recent generalizations of some of these results to RCD spaces (recall Remark 2.2(c)). b) Among functional inequalities, the relation between the Poincar´e inequality (spectral gap) and the diameter of Riemannian manifolds has been well studied (see [BBG, Be, Che, Cr]). In the Euclidean setting, quantitative estimates are studied in comparison with the Gaussian spaces in [DF, CF] for the Poincar´e inequality and in [BGRS, FIL, CF] for the logarithmic Sobolev inequality.

7 Quantitative isoperimetric inequality

This is the most technical step in this section, and the structure of the proof differs from that of [CMM, Proposition 6.4] because the diameter M is not bounded and the needles can be infinitely long (cf. e.g. , [CMM, Proposition 5.1, Corollary 5.4]). Proposition 7.3 (u is almost a center on most needles) If δ(A) is sufficiently small, then there exists a measurable set Qc⊂Q such that ν(Qc)≥1−δ(A)(1−ε)/(9 − 3ε) and. Instead of beads in [CMM], we use arrays of sublevel and superlevel of the guide function.

Therefore, the novelty of Theorem 7.5 lies in the construction of u as the leading function of the needle decomposition. This is a similar phenomenon to the rigidity of the Bakry-Ledoux isoperimetric inequality (under Ric∞ ≥ K > 0) in Theorem 2.8, where the conducting function plays a similar role to the Busemann function (see [Ma2] for details) . Returning to our quantitative research, the leading function u shares several properties with the Busemann function: u is 1-Lipschitz, most needles are long in both directions (limδ→0S =−∞ . and limδ→0T =∞in Proposition3.2 ), and the direction of most needles is the same (Theorem 7.4).

When, for example, some needle is a straight line, one can relate the associated Busemann function to the accompaniment function and obtain (7.8) in terms of that Busemann function. In this direction, one could also expect an 'almost splitting theorem' if metric measure spaces, namely (M, g,m) are close to the product space (R,| · |,γ)×Y in some sense (even when there is no infinity needle). This is an interesting and challenging problem, let us recall that Gromov's precompactness theorem ([Gr, §5.A]) does not apply under Ric∞ ≥K >0. b) Compared to the Cheeger–Gromoll type splitting theorem under Ric∞ ≥ 0 in.

In the splitting theorem, we claim that the space splits from the real line fitted with the Lebesgue measure, and so an upper bound on Ψ is needed to rule out Gaussian spaces (and hyperbolic spaces with highly convex weight functions). In the non-reversible case, however, the needle decomposition does not give the sharp isoperimetric inequality, and it is unclear whether one can generalize Theorem 7.5. See [Oh3] for more details on the non-reversible situation, and [Oh4] for a derivation of the sharp Bakry-Ledoux isoperimetric inequality for non-reversible Finsler manifolds.

The condition θ ̸= 1/2 was only used in Proposition 7.4, where we showed that one of Q−ℓ and Q+ℓ has a small volume. If this step is established in a different way, then all the other steps in the proof work and we can obtain Theorem 7.5 for θ = 1/2. We thank Fabio Cavalletti and Max Fathi for discussions during the "Geometry and Probability" workshop in Osaka (2019).

We refer to [AM] for the Bakry-Ledoux isoperimetric inequality on RCD(1,∞) spaces. e) There are two open problems related to Theorem 7.5. Our estimate δ(1−ε)/(9−3ε) does not seem optimal at all and, compared to the case of Gaussian spaces (recall (1.1)), the optimal order is probably√. g) Inspired by [DF, CF] we expect the forward measure u∗m to be close to γ ​​at the Wasserstein distance W1 or W2 above R. Mondino, Sharp and rigid isoperimetric inequalities in metric measure spaces with lower Ricci curvature bounds.

Sturm, On the equivalence of the entropy curvature condition and the Bochner inequality on metric gauge spaces. Gigli, Review of the proof of the splitting theorem in spaces with non-negative Ricci curvature. Milman, Beyond the traditional curvature dimension I: new model spaces for isoperimetric and concentration inequalities in the negative dimension.

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