arXiv subjects

Franz E. Schuster

Publications and source records attributed to Franz E. Schuster.

At least 19 recordsLinked to original sources

arXiv2024

Affine Quermassintegrals and Even Minkowski Valuations

It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton distribution. In case of a non-negative distribution, the polar volume of the associated Minkowski valuation gives rise to an isoperimetric inequality which strengthens the classical relation between the $i$th quermassintegral and the volume. This large family of inequalities unifies earlier results obtained for $i = 1$ and $n - 1$. In these cases, isoperimetric inequalities for affine quermassintegrals, specifically the Blaschke-Santaló inequality for $i = 1$ and the Petty projection inequality for $i = n - 1$, were proven to be the strongest inequalities. An analogous result for the intermediate degrees is established here. Finally, a new sufficient condition for the existence of maximizers for the polar volume of Minkowski valuations intertwining rigid motions reveals unexpected examples of volume inequalities having asymmetric extremizers.

math.MG↗

arXiv2021

Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one - the Blaschke-Santaló inequality - turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes the functional Blaschke-Santaló inequality.

math.MG↗

arXiv2021

Fixed points of Minkowski valuations

It is shown that for any sufficiently regular even Minkowski valuation $Φ$ which is homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are the only solutions to the fixed-point problem $Φ^2 K = αK$. This significantly generalizes results by Ivaki for projection bodies and suggests, via the Lutwak--Schneider class reduction technique, a new approach to Petty's conjectured projection inequality.

math.MG↗

arXiv2019

Sharp Sobolev inequalities via projection averages

A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$ Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine $L^p$ Sobolev inequality of Lutwak, Yang, and Zhang. When $p = 1$, the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.

math.FA↗

arXiv2019

Lutwak-Petty projection inequalities for Minkowski valuations and their duals

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.

math.MG↗

arXiv2019

Spherical centroid bodies

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.

math.MG↗

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↗

arXiv2016

Projection functions, area measures and the Alesker-Fourier transform

Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are constructed.

math.MG↗

arXiv2015

Minkowski Valuations and Generalized Valuations

A convolution representation of continuous translation invariant and SO(n) equivariant Minkowski valuations is established. This is based on a new classification of translation invariant generalized spherical valuations. As applications, Crofton and kinematic formulas for Minkowski valuations are obtained.

math.MG↗

arXiv2015

A characterization of Blaschke addition

A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of Lévy-Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.

math.MG↗

arXiv2014

Log-Concavity Properties of Minkowski Valuations

New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new classification of generalized translation invariant valuations.

math.MG↗

arXiv2014

Even Minkowski Valuations

A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.

math.MG↗

arXiv2014

Binary Operations in Spherical Convex Geometry

Characterizations of binary operations between convex bodies on the Euclidean unit sphere are established. The main result shows that the convex hull is essentially the only non-trivial projection covariant operation between pairs of convex bodies contained in open hemispheres. Moreover, it is proved that any continuous and projection covariant binary operation between all proper spherical convex bodies must be trivial.

math.MG↗

arXiv2012

An arithmetic proof of John's ellipsoid theorem

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of the corresponding 'easy part'.

math.MG↗

arXiv2012

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.

math.MG↗

arXiv2012

GL(n) contravariant Minkowski valuations

A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.

math.MG↗

arXiv2012

The Sine Transform of Isotropic Measures

Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projections.

math.MG↗

arXiv2012

The Steiner Formula for Minkowski Valuations

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations.

math.MG↗