arXiv subjects

Dmitry Faifman

Publications and source records attributed to Dmitry Faifman.

At least 19 recordsLinked to original sources

arXiv2026

The Faber-Krahn position of convex bodies and Gaussian measure inequalities

We say that a convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. We prove that this position is unique up to orthogonal transformations, answering a question of Schmuckenschlaeger from 2011. This is a corollary of a new log-convexity property of the first eigenvalue under positive definite linear deformations. While the centrally symmetric case follows from the Gaussian B-theorem, the extension to arbitrary convex bodies requires a quantitative analysis of conditioned Brownian motion. As consequences, we obtain a new proof of the Polya-Szego theorem for triangles and its analogue for simplices, and show that regular polygons minimize the first eigenvalue within their linear orbits of fixed volume. We also prove related convexity results for the first eigenvalue of the Ornstein-Uhlenbeck operator, the inverse inradius and the planar Cheeger constant. In a different direction, we show using similar ideas that the Gaussian conjugate Rogers-Shephard inequality due to Milman-Nakamura-Tsuji yields improved Schmuckenschlaeger-type bounds for intersections and Minkowski sums of centrally symmetric convex bodies.

math.SP↗

arXiv2026

The volume of tubes in Lie groups

The problem of computing the volume of tubes in riemannian manifolds goes back to Weyl and Hotelling. Here we find explicit Taylor series for the volume of a tube in a Lie group equipped with a bi-invariant metric. The coefficients are smooth valuations, given by the convolution powers of the surface area valuation. We show that the tube coefficients can be naturally described as the unique valuations given by universal formulas through the formalism of differential graded Lie and Gerstenhaber algebras; in fact, they are generated by the gauge action on the Maurer--Cartan cone in the free differential graded Lie algebra on one generator. Moreover, we introduce a new convolution product on the corresponding free Gerstenhaber algebra which is compatible with the convolution of valuations and differential forms. To complete the picture, we show that a Lie group -- not necessarily connected -- admits a smooth bi-invariant valuation, beyond the Euler characteristic and the Haar measure, if and only if it admits a bi-invariant riemannian metric.

math.DG↗

arXiv2025

Invariant valuations on Lie groups

Convolution of valuations was introduced by the first named author and Fu for linear spaces, and later by Alesker and the first named author for compact Lie groups. In this paper we study the convolution of invariant valuations on Lie groups. First, we obtain an explicit formula for the convolution of left-invariant valuations on compact groups in terms of differential forms. Independently, we show that a connected Lie group admits smooth bi-invariant valuations beyond the Euler characteristic and the Haar measure if and only if the group is the product of a compact group and a linear space. Finally, we use these two results to define the convolution of bi-invariant smooth valuations on an arbitrary unimodular Lie group, thus unifying both previously defined convolution operations.

math.DG↗

arXiv2024

Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces

The Banach isometric conjecture asserts that a normed space with all of its $k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an elegant topological argument. We refine their method to arrive at a stable version of their result: if all $2$-dimensional subspaces are almost isometric, then the space is almost Euclidean. Furthermore, we show that a $2$-dimensional surface, which is not a torus or a Klein bottle, equipped with a near-monochromatic Finsler metric, is approximately Riemannian. The stability is quantified explicitly using the Banach-Mazur distance.

math.MG↗

arXiv2023

The Fourier transform on valuations is the Fourier transform

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker--Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time.

math.CA↗

arXiv2023

Volume growth of Funk geometry and the flags of polytopes

We consider the Holmes--Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santal\'o point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon.

math.MG↗

arXiv2023

Convex valuations, from Whitney to Nash

We consider the Whitney problem for valuations: does a smooth $j$-homogeneous translation-invariant valuation on $\mathbb R^n$ exist that has given restrictions to a fixed family $S$ of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for $S=\mathrm{Gr}_r(\mathbb R^n)$, the grassmannian of $r$-planes, this condition becomes sufficient once $r\geq j+2$. This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension $2$. We then look for conditions on $S$ when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.

math.DG↗

arXiv2022

Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

We investigate the support of a distribution $f$ on the real grassmannian $\mathrm{Gr}_k(\mathbb R^n)$ whose spectrum, namely its nontrivial $\mathrm O(n)$-components, is restricted to a subset $\Lambda$ of all $\mathrm O(n)$-types. We prove that unless $\Lambda$ is co-sparse, $f$ cannot be supported at a point. We utilize this uncertainty principle to prove that if $2\leq k\leq n-2$, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell $\Sigma^k$. The same holds for certain more general $\alpha$-cosine transforms and for the Radon transform between grassmannians, and more generally for various $\mathrm{GL}_n(\mathbb R)$-modules. These results are then applied to convex geometry and geometric tomography, where sharper versions of the Aleksandrov projection theorem, Funk section theorem, and Klain's and Schneider's injectivity theorems for convex valuations are obtained.

math.RT↗

arXiv2021

Crofton formulas in pseudo-Riemannian space forms

Crofton formulas on simply-connected Riemannian space forms allow to compute the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners, by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

math.DG↗

arXiv2020

A Funk perspective on billiards, projective geometry and Mahler volume

We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are invariant under projective transformations as well as projective duality. These include the Holmes-Thompson volume and surface area of convex subsets, and the length spectrum of their boundary, extending results of Holmes-Thompson and \'Alvarez Paiva on Sch\"affer's dual girth conjecture. We explore in particular Funk billiards, which generalize hyperbolic billiards in the same way that Minkowski billiards generalize Euclidean ones, and extend a result of Gutkin-Tabachnikov on the duality of Minkowski billiards. We next consider the volume of outward balls in Funk geometry. We conjecture a general affine inequality corresponding to the volume maximizers, which includes the Blaschke-Santal\'o and centro-affine isoperimetric inequalities as limit cases, and prove it for unconditional bodies, yielding a new proof of the volume entropy conjecture for the Hilbert metric for unconditional bodies. As a by-product, we obtain generalizations to higher moments of inequalities of Ball and Huang-Li, which in turn strengthen the Blaschke-Santal\'o inequality for unconditional bodies. Lastly, we introduce a regularization of the total volume of a smooth strictly convex 2-dimensional set equipped with the Funk metric, resembling the O'Hara M\"obius energy of a knot, and show that it is a projective invariant of the convex body.

math.DG↗

arXiv2020

Uniqueness of curvature measures in pseudo-Riemannian geometry

The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a K\"unneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

math.DG↗

arXiv2019

The Weyl principle on the Finsler frontier

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. As a consequence, the intrinsic volumes behave naturally under isometric immersions. This phenomenon, subsequently observed in a number of different geometric settings, is commonly referred to as the Weyl principle. In general normed spaces, the Holmes-Thompson intrinsic volumes naturally extend the Euclidean intrinsic volumes. The purpose of this note is to investigate the applicability of the Weyl principle to Finsler manifolds. We show that while in general the Weyl principle fails, a weak form of the principle unexpectedly persists in certain settings.

math.DG↗

arXiv2019

Curvature Measures of Pseudo-Riemannian Manifolds

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz-Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern-Gauss-Bonnet integrand. Finally, we deduce a Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.

math.DG↗

arXiv2017

Contact integral geometry and the Heisenberg algebra

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure. Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle. Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere. Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.

math.DG↗

arXiv2017

Crofton Formulas and Indefinite Signature

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the $(p-1)$ homogeneous case of $O(p,p)$, yielding a Crofton formula for the centro-affine surface area when $p\not\equiv 3\mod 4$. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all $O(p,2)$-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the $|\det X|^s$ family of distributions on the space of symmetric matrices, which we use to construct a family of $O(p,q)$-invariant Crofton distributions. We conjecture there are no others, which we then prove for $O(p,2)$ with $p$ even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

math.DG↗

arXiv2016

Valuation theory of indefinite orthogonal groups

Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain-Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.

math.DG↗

arXiv2015

Generalized translation invariant valuations and the polytope algebra

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.

math.DG↗

arXiv2014

Lipschitz functions on the infinite-dimensional torus

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that $f$ is a measurable, real-valued, Lipschitz function on the torus $\mathbb{T}^{\infty}$. We prove that there exists a number $a \in \mathbb R$ with the following property: For any $ε> 0$ there exists a parallel, infinite-dimensional subtorus $M \subseteq \mathbb T^{\infty}$ such that the restriction of the function $f-a$ to the subtorus $M$ has an $L^{\infty}(M)$-norm of at most $ε$.

math.PR↗