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Carsten Schütt

Publications and source records attributed to Carsten Schütt.

18 recordsLinked to original sources
arXiv2026

Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. Böröczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$.

math.MG↗
arXiv2026

Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

We study dual volume approximation of the Euclidean ball by polytopes with a prescribed number of $k$-dimensional faces. This continues the authors' previous work on intrinsic volume approximation of the ball by polytopes with a fixed number of $k$-faces, and develops the corresponding dual radial theory. Our first main result gives a nonasymptotic lower bound for the volume deficit of an inscribed polytope $P_M\subset B_d$ with at most $M$ $k$-faces, for $0\leq k\leq \lfloor d/2\rfloor$. The estimate has the form \[\operatorname{vol}_d(B_d\setminus P_M) \geq \frac{1-e^{-1}}{2}κ_d\min\left\{ 1,\frac{d}{2}\left(\frac{ω_d}{4κ_{d-1}}\right)^{\frac{2}{d-1}} M^{-\frac{2}{d-1}}\right\},\] and, in the vertices case $k=0$, in the large-$M$ regime it recovers the order of the lower bound of Gordon, Reisner and Schütt (J. Approx. Theory, 1997). We also prove the polar counterpart for the mean width excess of circumscribed polytopes with at most $M$ $k$-faces, for $\lceil d/2\rceil-1\leq k\leq d-1$. More generally, using the analytic extension of the dual volume deviations introduced by Besau, Hoehner and Kur (Int. Math. Res. Not., 2021), we obtain nonasymptotic lower bounds for all $q\in\mathbb R$, including $q=0$, in both the inscribed and circumscribed models.

math.MG↗
arXiv2026

Illumination bodies on Riemannian manifolds

We prove a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body, which holds on Riemannian manifolds with Ricci curvature bounded from below. The $δ$-illumination body of a subset of a Riemannian manifold is defined to be the set of all points such that the union of all minimizing geodesic segments joining the point to the set has volume at most $δ$.

math.DG↗
arXiv2025

Expected extremal area of facets of random polytopes

We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in $\mathbb{R}^n$. The extremal properties of interest are the expected values of the maximum and minimum surface area among facets. We determine the asymptotic growth in every fixed dimension, up to absolute constants.

math.PR↗
arXiv2024

Random approximation of convex bodies in Hausdorff metric

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant $K_n$ is given by the convex hull of $n$ independent random points chosen uniformly on the boundary or in the interior of $K$. When $K$ is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as $n\to\infty$. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.

math.MG↗
arXiv2024

Weighted floating functions and weighted functional affine surface areas

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as to the classical affine surface areas for convex bodies is discussed in detail.

math.MG↗
arXiv2022

Spherical convex hull of random points on a wedge

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 dimension $d-2$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on $\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on $\mathbb{S}_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

math.PR↗
arXiv2021

Affine invariant maps for log-concave functions

Affine invariant points and maps for sets were introduced by Grünbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show that among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santaló point. We also show that the recently introduced floating functions and the John- and Löwner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.

math.FA↗
arXiv2018

Flag numbers and floating bodies

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 results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.

math.MG↗
arXiv2016

On the geometry of projective tensor products

In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_π\ell_q^n\otimes_π\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ and $q=2$. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype $2$ constant, we obtain information on the cotype of these $3$-fold projective tensor products. Our results naturally generalize to $k$-fold products $\ell_{p_1}^n\otimes_π\dots \otimes_π\ell_{p_k}^n$ with $k\in\mathbb N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$.

math.FA↗
arXiv2010

A note on Mahler's conjecture

Let $K$ be a convex body in $\mathbb{R}^n$ with Santaló point at 0\. We show that if $K$ has a point on the boundary with positive generalized Gauß curvature, then the volume product $|K| |K^\circ|$ is not minimal. This means that a body with minimal volume product has Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

math.FA↗
arXiv2008

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics of sequences of independent identically distributed random variables with log-concave distribution in terms of Orlicz norms associated with the distribution function of the random variables.

math.PR↗
arXiv1996

Floating body, illumination body, and polytopal approximation

Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\ vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and $Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional faces. Then $$ vol_{d}(K^{t} \setminus K) \leq cd^{4} vol_{d}(Q_{n} \setminus K) $$ where $$ n \leq \frac{c}{dt} \ vol_{d}(K^{t} \setminus K) $$

math.MG↗
arXiv1996

Umbrellas and polytopal approximation of the euclidean ball

There are two positive, absolute constants $c_{1}$ and $c_{2}$ so that the volume of the difference set of the $d$-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than $$ c_{2}\ d\ {n}^{-\frac{2}{d-1}}vol_d(B^d_2) $$ for $n \geq (c_{1}\ d)^{\frac{d-1}{2}}$.

math.MG↗
arXiv1994

On the embedding of 2-concave Orlicz spaces into $L^1$

In [K--S 1] it was shown that $$ \underset π \to {\text{Ave}} (\sum_{i=1}^{n}|x_i a_{π(i)}|^2)^{\frac {1}{2}} $$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1, a_2,....,a_n$ so that the above expression is equivalent to a given Orlicz norm.

math.FA↗
arXiv1993

Random polytopes and affine surface area

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with $C^3$ boundary and everywhere positive curvature $$ c(d)\lim_{n \to \infty} \frac {vol_d(K)-\Bbb E(K,n)}{(\frac{vol_d(K)}{n})^{\frac{2}{d+1}}} =\int_{\partial K} κ(x)^{\frac{1}{d+1}}dμ(x) $$ where $κ(x)$ denotes the Gauß-Kronecker curvature. We show that the same formula holds for all convex bodies if $κ(x)$ denotes the generalized Gauß-Kronecker curvature.

math.MG↗