arXiv subjects

Emanuel Milman

Publications and source records attributed to Emanuel Milman.

At least 19 recordsLinked to original sources

arXiv2026

Thin-Shell implies small-ball deviation via Gaussian tilts

We show that uniform thin-shell estimates for isotropic log-concave measures $\mu$ on $\mathbb{R}^n$ yield precise and explicit deviation estimates for the Euclidean norm $|X|$ below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: $$ \mu\left(|X|\le \sqrt{n}-s\right) \le \exp\left\{ -\frac{s^2}{4} \left(1+O\left(\frac{s}{\sqrt{n}}\right)\right) \right\} \qquad \forall\, s\in(0,\sqrt{n}). $$ In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when $\mu$ has general covariance structure) of these deviation estimates is also obtained.

math.FA↗

arXiv2026

Spectral and Isoperimetric Bounds on Flat Tori

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda \subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $\xi$ in the dual lattice $\Lambda^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K \xi,\xi\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ (equivalently, the length of the shortest nonzero dual vector $\lambda_1(\Lambda^*)$) in terms of the maximal covariance of $K$: \[ \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda) = 4\pi^2 \lambda_1(\Lambda^*)^2 \geq \frac{\pi^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_\Lambda$ of a lattice with $\det \Lambda = 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\quad \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\quad \lambda_1(\Lambda^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lov\'asz--Simonovits conjecture for all flat tori. While there are lattices $\Lambda$ and corresponding Voronoi cells $K = K_\Lambda$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).

math.SP↗

arXiv2026

The Gaussian Conjugate Rogers-Shephard Inequality

We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $\gamma$ on $\mathbb{R}^n$, stating that \[ \gamma(K) \gamma(L) \leq \gamma(K\cap L) \gamma(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances.

math.FA↗

arXiv2025

On the connectedness of a minimizing cluster's boundary

We verify that an isoperimetric minimizing cluster on a simply connected homogeneous Riemannian manifold with at most one end always has connected boundary. In particular, the boundary of a single-bubble isoperimetric minimizer on such manifolds must be connected, and hence all isoperimetric sets and their complements must be connected. This is demonstrably false without the simple connectedness assumption or the restriction on the number of ends.

math.DG↗

arXiv2025

Multi-Bubble Isoperimetric Problems

We survey recent advancements in the characterization of multi-bubble isoperimetric minimizers and the stability of soap bubble partitions. We conclude with some related open problems.

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arXiv2025

Standard bubbles (and other M\"obius-flat partitions) on model spaces are stable

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In fact, stability holds for all standard $\textit{partitions}$, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in $\mathbb{G}^n$ ($n\geq 2$) obeying Plateau's laws and whose interfaces are all $\textit{flat}$, is stable. Our results apply to non-standard partitions as well - starting with any (regular) flat Voronoi partition in $\mathbb{S}^n$ and applying M\"obius transformations and stereographic projections, the resulting partitions in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$ are stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential.

math.DG↗

arXiv2025

Gaussian Correlation via Inverse Brascamp-Lieb

We give a simple alternative proof of Royen's Gaussian Correlation inequality by using (a slightly generalized version of) Nakamura-Tsuji's symmetric inverse Brascamp-Lieb inequality for even log-concave functions. We explain that this inverse inequality is in a certain sense a dual counterpart to the forward inequality of Bennett-Carbery-Christ-Tao and Valdimarsson, and that the log-concavity assumption therein cannot be omitted in general.

math.FA↗

arXiv2024

A Blaschke-Santal\'o inequality for unconditional log-concave measures

The Blaschke-Santal\'o inequality states that the volume product $|K| \cdot |K^{o}|$ of a symmetric convex body $K \subset \mathbb{R}^n$ is maximized by the standard Euclidean unit-ball. Cordero-Erausquin asked whether the inequality remains true for all even log-concave measures. We briefly survey the literature around this question and provide details for the known fact that the inequality holds true for all unconditional log-concave measures.

math.FA↗

arXiv2024

Fixed and Periodic Points of the Intersection Body Operator

The intersection body $IK$ of a star-body $K$ in $\mathbb{R}^n$ was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when $n \geq 3$, $I^2 K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ iff $K$ is a centered Euclidean ball, answering long-standing questions by Lutwak, Gardner, and Fish-Nazarov-Ryabogin-Zvavitch. To this end, we recast the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$.

math.FA↗

arXiv2024

Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs

We study isoperimetric inequalities on "slabs", namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension-one base. As our two main applications, we consider the case when the base is the flat torus $\mathbb{R}^2 / 2 \mathbb{Z}^2$ and the standard Gaussian measure in $\mathbb{R}^{n-1}$. The isoperimetric conjecture on the three-dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to $0$, $1/2$ and $1$ by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths $(\beta,1,1)$ in a new range of relatives volumes $\bar v \in [0,1/2]$. In particular, we confirm the conjecture for the standard cube ($\beta=1$) for all $\bar v \leq 0.120582$, when $\beta \leq 0.919431$ for the entire range where spheres are conjectured to be minimizing, and also for all $\bar v \in [0,1/2] \setminus (\frac{1}{\pi} - \frac{\beta}{4},\frac{1}{\pi} + \frac{\beta}{4})$. When $\beta \leq 0.919431$ we reduce the validity of the full conjecture to establishing that the half-plane $\{ x \in [0,\beta] \times [0,1]^2 \; ; \; x_3 \leq \frac{1}{\pi} \}$ is an isoperimetric minimizer. We also show that the analogous conjecture on a high-dimensional cube $[0,1]^n$ is false for $n \geq 10$. In the case of a slab with a Gaussian base of width $T>0$, we identify a phase transition when $T = \sqrt{2 \pi}$ and when $T = \pi$. In particular, while products of half-planes with $[0,T]$ are always minimizing when $T \leq \sqrt{2 \pi}$, when $T > \pi$ they are never minimizing, being beaten by Gaussian unduloids. In the range $T \in (\sqrt{2 \pi},\pi]$, a potential trichotomy occurs.

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arXiv2023

$L^p$-Minkowski Problem under Curvature Pinching

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$, is comparable to the standard Euclidean metric of $\mathbb{R}^{n}$ up-to a factor of $\gamma > 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_\gamma := 1 - \frac{n+1}{\gamma}$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma < \infty$. In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.

math.DG↗

arXiv2023

Plateau Bubbles and the Quintuple Bubble Theorem on $\mathbb{S}^n$

Sullivan's multi-bubble isoperimetric conjectures in $n$-dimensional Euclidean and spherical spaces assert that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$. The double-bubble conjecture on $\mathbb{R}^3$ was confirmed by Hutchings-Morgan-Ritor\'e-Ros (and later extended to $\mathbb{R}^n$). The double-bubble conjecture on $\mathbb{S}^n$ ($n \geq 2$) and the triple- and quadruple- bubble conjectures on $\mathbb{R}^n$ and $\mathbb{S}^n$ (for $n \geq 3$ and $n \geq 4$, respectively) were recently confirmed in our previous work, but the approach employed there does not seem to allow extending these results further. In this work, we confirm the quintuple-bubble conjecture on $\mathbb{S}^n$ ($n \geq 5$), and as a consequence, by approximation, also the quintuple-bubble conjecture on $\mathbb{R}^n$ ($n \geq 5$) but without the uniqueness assertion. Moreover, we resolve the conjectures on $\mathbb{S}^n$ and on $\mathbb{R}^n$ (without uniqueness) for all $q \leq n+1$, conditioned on the assumption that the singularities which appear at the meeting locus of several bubbles obey a higher-dimensional analogue of Plateau's laws. Another scenario we can deal with is when the bubbles are full-dimensional ("in general position"), or arrange in some good lower-dimensional configurations. To this end, we develop the spectral theory of the corresponding Jacobi operator (finding analogies with the quantum-graph formalism), and a new method for deforming the bubbles into a favorable configuration. As a by-product, we show that the Jacobi operator on a minimizing configuration always has index precisely $q-1$ and hence the corresponding isoperimetric profile is concave, answering a question of Heppes. Several compelling conjectures are proposed, which would allow extending our results to all $q \leq n+1$ unconditionally.

math.DG↗

arXiv2023

Uniqueness of solutions to a class of isotropic curvature problems

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various non-linearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems.

math.DG↗

arXiv2023

Centro-Affine Differential Geometry and the Log-Minkowski Problem

We interpret the log-Brunn-Minkowski conjecture of Böröczky-Lutwak-Yang-Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert-Brunn-Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $\mathbb{R}^n$ is a centro-affine unit-sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn-Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $\bar K$ in $\mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $\bar K \subset K \subset 8 \bar K$, so that $K$ satisfies the log-Minkowski conjectured inequality, and so that $K$ is uniquely determined by its cone-volume measure $V_K$. If $\bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+ε$, is obtained as well.

math.FA↗

arXiv2022

Sharp Isoperimetric Inequalities for Affine Quermassintegrals

The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke-Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of $k=1,\ldots,n-1$, in a single unified framework. In fact, it turns out that ellipsoids are the only local minimizers with respect to the Hausdorff topology. For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty's inequality is interpreted as an integrated form of a generalized Blaschke--Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general $L^p$-moment quermassintegrals, and interpret the case $p=0$ as a sharp averaged Loomis--Whitney isoperimetric inequality.

math.MG↗

arXiv2022

A sharp centro-affine isospectral inequality of Szegö--Weinberger type and the $L^p$-Minkowski problem

We establish a sharp upper-bound for the first non-zero even eigenvalue (corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski operator associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affine invariant, attaining equality if and only if $K$ is a (centered) ellipsoid; this is reminiscent of the (non affine invariant) classical Szegö--Weinberger isospectral inequality for the Neumann Laplacian. The new upper-bound complements the conjectural lower-bound, which has been shown to be equivalent to the log-Brunn-Minkowski inequality and is intimately related to the uniqueness question in the even log-Minkowski problem. As applications, we obtain new strong non-uniqueness results in the even $L^p$-Minkowski problem in the subcritical range $-n < p < 0$, as well as new rigidity results for the critical exponent $p=-n$ and supercritical regime $p < -n$. In particular, we show that any $K$ as above which is not an ellipsoid is a witness to non-uniqueness in the even $L^p$-Minkowski problem for all $p \in (-n,p_K)$ and some $p_K \in (-n,0)$, and that $K$ can be chosen so that $p_K$ is arbitrarily close to $0$.

math.FA↗

arXiv2022

The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$

The multi-bubble isoperimetric conjecture in $n$-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$. The double-bubble conjecture on $\mathbb{R}^3$ was confirmed in 2000 by Hutchings-Morgan-Ritor\'e-Ros, and is nowadays fully resolved for all $n \geq 2$. The double-bubble conjecture on $\mathbb{S}^2$ and triple-bubble conjecture on $\mathbb{R}^2$ have also been resolved, but all other cases are in general open. We confirm the conjecture on $\mathbb{R}^n$ and on $\mathbb{S}^n$ for all $q \leq \min(5,n+1)$, namely: the double-bubble conjectures for $n \geq 2$, the triple-bubble conjectures for $n \geq 3$ and the quadruple-bubble conjectures for $n \geq 4$. In fact, we show that for all $q \leq n+1$, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to $\mathbb{S}^n$, its cells are obtained as the Voronoi cells of $q$ affine-functions, or equivalently, as the intersection with $\mathbb{S}^n$ of convex polyhedra in $\mathbb{R}^{n+1}$. Moreover, the cells (including the unbounded one) are necessarily connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes. We also show for all $q \leq n+1$ that a minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble. The proof makes crucial use of considering $\mathbb{R}^n$ and $\mathbb{S}^n$ in tandem and of M\"obius geometry and conformal Killing fields; it does not rely on establishing a PDI for the isoperimetric profile as in the Gaussian setting, which seems out of reach in the present one.

math.DG↗

arXiv2021

The Gaussian Double-Bubble and Multi-Bubble Conjectures

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). In particular, the case $q=3$ confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ ($n \geq 2$) into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ angles (forming a tripod or "Y" shape in the plane). The case $q=2$ recovers the classical Gaussian isoperimetric inequality. To establish the Multi-Bubble conjecture, we show that in the above range of $q$, stable regular clusters must have flat interfaces, therefore consisting of convex polyhedral cells (with at most $q-1$ facets). In the Double-Bubble case $q=3$, it is possible to avoid establishing flatness of the interfaces by invoking a certain dichotomy on the structure of stable clusters, yielding a simplified argument.

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