Florian Besau

TU Wien | TU Wien · Institute of Discrete Mathematics and Geometry

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santal\'o inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume...

Consider two half-spaces $$H_1^+$$ H 1 + and $$H_2^+$$ H 2 + in $${\mathbb {R}}^{d+1}$$ R d + 1 whose bounding hyperplanes $$H_1$$ H 1 and $$H_2$$ H 2 are orthogonal and pass through the origin. The intersection $${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ S 2 , + d : = S d ∩ H 1 + ∩ H 2 + is a spherical convex subset of the d -di...

A new intrinsic volume metric is introduced for the class of convex bodies in [Formula: see text]. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and s...

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows...

Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$, which contains a great subsphere of dimensio...

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hu...

Central limit theorems for the log-volume of a class of random convex bodies in R n \mathbb {R}^n are obtained in the high-dimensional regime, that is, as n → ∞ n\to \infty . In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating ve...

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hu...

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high d...

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak's dual Brunn--Minkowski theory are established. The results follow from a central l...

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are...

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high...

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high...

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, whi...

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic resul...

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic resul...

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to deriv...

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation be...

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area turns out to be a spherical a...

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 bin...