arXiv subjects

Jan Kotrbatý

Publications and source records attributed to Jan Kotrbatý.

11 recordsLinked to original sources

arXiv2026

Around higher-order Godbersen conjectures

We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to Fáry and Rédei.

math.MG↗

arXiv2026

Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Our main result establishes the hard Lefschetz theorem and the Hodge-Riemann relations in full generality. As a consequence, we obtain McMullen's quadratic inequalities, which are valid for strongly isomorphic polytopes and known to fail in general, for convex bodies with smooth and strictly positively curved boundary. Our proof is based on elliptic operator theory and on perturbation theory applied to unbounded operators on a natural Hilbert space completion of the space of smooth translation-invariant valuations.

math.DG↗

arXiv2026

Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality

We prove that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the classical Rogers-Shephard inequality. We also prove that simplices are the only extremizers among convex polytopes. Finally, we use this inequality to prove an $L_p$-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.

math.MG↗

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↗

arXiv2025

On a generalization of Godbersen's conjecture

The long-standing Godbersen's conjecture asserts that the Rogers-Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture is known in several special cases, notably for anti-blocking convex bodies. In this note, we propose a generalization of Godbersen's conjecture that refines Schneider's generalization of the Rogers-Shephard inequality to higher-order difference bodies and prove our conjecture for anti-blocking convex bodies. Moreover, we relate the conjectured inequality to the higher-rank mixed volume defined by the author and Wannerer which leads to an equivalent formulation in terms of the Alesker product of smooth, translation invariant valuations.

math.MG↗

arXiv2022

Integral geometry on the octonionic plane

We describe explicitly the algebra of Spin(9)-invariant, translation-invariant, continuous valuations on the octonionic plane. Namely, we present a basis in terms of invariant differential forms and determine the Bernig-Fu convolution on this space. The main technical ingredient we introduce is an extension of the invariant theory of the Lie group Spin(7) to the isotropy representation of the action of Spin(9) on the 15-dimensional sphere, reflecting the underlying octonionic structure. As an application, we compute the principal kinematic formula on the octonionic plane and express in our basis certain Spin(9)-invariant valuations introduced previously by Alesker.

math.MG↗

arXiv2022

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies

The Alesker-Bernig-Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker-Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge-Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge-Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.

math.DG↗

arXiv2021

On Hodge-Riemann relations for translation-invariant valuations

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and the hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for the Alesker algebra is conjectured, and the conjecture is proved in two particular situations: for even valuations, and for 1-homogeneous valuations. The latter result is then used to deduce a special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lefschetz theorem and of the Hodge-Riemann relations are conjectured, and it is shown that the Aleksandrov-Fenchel inequality follows from the latter in its full generality.

math.MG↗

arXiv2021

On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities

A version of the Hodge-Riemann relations for valuations was recently conjectured and proved in several special cases by the first-named author. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one the Hodge-Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge-Riemann relations for the convolution directly imply the Aleksandrov-Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov-Fenchel inequality for lower dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by S. Alesker

math.MG↗

arXiv2019

Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a $Spin(9)$-structure

It is well known that there is a unique $Spin(9)$-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak $Spin(9)$-structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the present article, a new explicit algebraic formula for the $Spin(9)$-invariant 8-form is given. The approach we use generalizes the standard expression of the Kähler 2-form. Namely, the invariant 8-form is constructed only from the two octonion-valued coordinate 1-forms on the octonionic plane. For completeness, analogous expressions for the Kraines form, the Cayley calibration and the associative calibration are also presented.

math.RT↗