arXiv subjects

Christoph Haberl

Publications and source records attributed to Christoph Haberl.

9 recordsLinked to original sources

arXiv2025

Exponential valuations on lattice polygons valued at formal power series

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The classification is done in terms of formal power series that satisfy certain functional equations. We align our classification with the decomposition into so-called dilative components.

math.MG↗

arXiv2025

Exponential valuations on lattice polygons

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.

math.NT↗

arXiv2018

Affine vs. Euclidean isoperimetric inequalities

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of Lp isoperimetric and Sobolev type inequalities are also established.

math.MG↗

arXiv2015

Centro-Affine Tensor Valuations

We completely classify all measurable $\operatorname{SL}(n)$-covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors. It is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from $L_p$ surface area measures. This generalizes and unifies earlier results for the scalar, vector and matrix valued case.

math.MG↗

arXiv2015

Moments and Valuations

All measurable and $\operatorname{SL}(n)$-covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified. The moment vector is shown to be essentially the only such valuation.

math.MG↗

arXiv2014

Valuations and Surface Area Measures

We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is SL(n) contravariant of degree one. Moreover, for all real $p$, an $L_p$ version of the above result is established for GL(n) contravariant valuations of degree $p$. This provides a characterization of the $L_p$ surface area measures from the $L_p$ Brunn-Minkowski theory.

math.MG↗

arXiv2013

The Centro-Affine Hadwiger Theorem

All upper semicontinuous and SL(n) invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.

math.MG↗

arXiv2009

An Asymmetric Affine Pólya--Szegö Principle

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on $\mathbb{R}^n$. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.

math.FA↗

arXiv2008

General Lp affine isoperimetric inequalities

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.

math.DG↗