arXiv subjects

Jonas Knoerr

Publications and source records attributed to Jonas Knoerr.

At least 19 recordsLinked to original sources

arXiv2026

Isometry invariant valuations on spherical polytopes

We show that every continuous and isometry invariant valuation on spherical polytopes is a linear combination of the spherical intrinsic volumes. The proof relies on a weak differentiability property satisfied by valuations on polytopes in $\mathbb{R}^n$ with a natural smoothness property with respect to the action of the affine group. This enables us to transfer several results established by Alesker for quasi-smooth valuations to the polytopal setting, and to reduce the problem to the translation invariant case of measurable valuations on polytopes.

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arXiv2026

Rigid motion invariant valuations on polytopes

We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for translation invariant, measurable, $1$-homogeneous, and simple valuations. As a consequence, we obtain a similar characterization of the Steiner point map on polytopes.

math.MG↗

arXiv2026

Translation invariant area measures on convex bodies

We introduce the space of continuous and translation invariant area measures, which are measure-valued functionals on the space of convex bodies satisfying a certain locality condition. Our main result shows that the space of $\mathrm{GL}(n,\mathbb{R})$-smooth area measures coincides with the space of area measures obtained by integration with respect to the normal cycle. We show how this result yields Hadwiger-type classification results for continuous area measures that are equivariant with respect to compact groups acting transitively on the unit sphere. In addition, we establish a general density criterion for invariant submodules and show that mixed area measures generate dense submodules with respect to suitable topologies on the space of continuous area measures. As a byproduct, we discuss how McMullen's Conjecture can be obtained directly from the representation of $\mathrm{GL}(n,\mathbb{R})$-smooth translation invariant valuations on convex bodies in terms of integration with respect to the normal cycle.

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arXiv2026

Integral representation of polynomial local functionals on convex functions

Integral representations for continuous polynomial local functionals on convex functions are established in terms of a finite family of polynomials. This result is obtained by approximation from a classification of the dense subspace of smooth polynomial local functionals, which is based on a Paley--Wiener--Schwartz-type classification of the Goodey--Weil distributions associated to these functionals under support restrictions. As an application, density results for various families of Monge--Amp\`ere-type operators are established.

math.FA↗

arXiv2025

Polynomial local functionals on convex functions

We show that every continuous local functional on the space of finite convex functions on $\mathbb{R}^n$ is a valuation. This relation is used to establish a homogeneous decomposition for the class of polynomial local functionals as well as a classification of translation or rigid motion invariant polynomial local functionals. In addition we discuss implications for the compact-open topology on the space of polynomial local functionals.

math.FA↗

arXiv2025

Localization of valuations and Alesker's irreducibility theorem

We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are representable by integration with respect to the normal cycle. This allows us to reduce the statement to a corresponding result for the representation of $\mathfrak{sl}(n)$ on the space of these differential forms.

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arXiv2025

A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions

Continuous dually epi-translation invariant valuations on convex functions are characterized in terms of the Fourier-Laplace transform of the associated Goodey-Weil distributions. This description is used to obtain integral representations of the smooth vectors of the natural representation of the group of translations on the space of these valuations. As an application, a complete classification of all closed and affine invariant subspaces is established, yielding density results for valuations defined in terms of mixed Monge-Amp\`ere operators.

math.FA↗

arXiv2024

Zonal valuations on convex bodies

A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence of these principal value integrals is obtained from a new weighted version of an inequality for the volume of spherical caps due to Firey. For Minkowski valuations, this implies a refinement of the convolution representation by Schuster and Wannerer in terms of singular integrals. As a further application, a new proof of the classification of $\mathrm{SO}(n)$-invariant, continuous, and dually epi-translation invariant valuations on the space of finite convex functions by Colesanti, Ludwig, and Mussnig is obtained.

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arXiv2024

Polynomial valuations on convex functions and their maximal extensions

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the support of these valuations. The results rely on a homogeneous decomposition for the space of polynomial valuations of bounded degree and the support properties of certain distributions associated to the homogeneous components. As an application, an explicit integral representation for valuations of top degree is established.

math.FA↗

arXiv2024

A geometric decomposition for unitarily invariant valuations on convex functions

Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these valuations decomposes into a direct sum of subspaces defined in terms of vanishing properties with respect to restrictions to a finite family of special subspaces of $\mathbb{C}^n$, mirroring the behavior of the hermitian intrinsic volumes introduced by Bernig and Fu. Unique representations of these valuations in terms of principal value integrals involving two families of Monge-Amp\`ere-type operators are established

math.FA↗

arXiv2024

Equivariant Valuations on Convex Functions

We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin.

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arXiv2024

The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere

We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace.

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arXiv2023

Smooth valuations on convex bodies and finite linear combinations of mixed volumes

It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, $k$-homogeneous valuation on convex bodies in $\mathbb{R}^n$ can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most $N_{n,k}$ summands, where $N_{n,k}$ is a constant depending on $n$ and $k$ only. Moreover, $n-k-1$ of the arguments of the mixed volumes can be chosen to be ellipsoids that do not depend on the valuation. The result is based on a corresponding description of smooth valuations in terms of finite linear combinations of mixed volumes.

math.MG↗

arXiv2023

Monge-Amp\`ere operators and valuations

Two classes of measure-valued valuations on convex functions related to Monge-Amp\`ere operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on $\mathbb{R}^n$ that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations of these valuations and a description in terms of mixed Monge-Amp\`ere operators are established, as well as a characterization of $\mathrm{SO}(n)$-equivariant valuations in terms of Hessian measures.

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arXiv2023

From valuations on convex bodies to convex functions

A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.

math.MG↗

arXiv2022

Singular valuations and the Hadwiger theorem on convex functions

We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the functional intrinsic volumes using integration over the differential cycle and provide a new representation of these functionals as principal value integrals with respect to the Hessian measures.

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arXiv2022

Equivariant Endomorphisms of Convex Functions

Characterizations of all continuous, additive and $\mathrm{GL}(n)$-equivariant endomorphisms of the space of convex functions on a Euclidean space $\mathbb{R}^n$, of the subspace of convex functions that are finite in a neighborhood of the origin, and of finite convex functions are established. Moreover, all continuous, additive, monotone endomorphisms of the same spaces, which are equivariant with respect to rotations and dilations, are characterized. Finally, all continuous, additive endomorphisms of the space of finite convex functions of one variable are characterized.

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arXiv2021

Unitarily invariant valuations on convex functions

Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge-Amp\`ere-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier-Laplace transform of the associated Goodey-Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of $\mathbb{C}^n$.

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