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Markus Kiderlen

Publications and source records attributed to Markus Kiderlen.

15 recordsLinked to original sources
arXiv2026

Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound

Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space. A set $B$ is called a barrier (or an `opaque set') for $K$ if every line that intersects $K$, also intersects $B$. Although this concept was introduced more than a century ago, the barrier with minimal surface area for a given set $K$ is still unknown, even in the two-dimensional case. A classical lower bound by Jones states that the surface area $S(B)$ of a sufficiently regular barrier $B$ is at least $S(\partial K)/2$, half the surface area of the boundary of $K$. We will extend a known stability version for $n=2$ to arbitrary dimensions: if $S(B)-S(\partial K)/2$ is small, then the orientation measure of $B$ is close to the surface area measure of a symmetrization of $K$. For instance, if $K$ is the unit cube in 3D, most of the points of a barrier with surface area close to $3$ must have almost axis parallel normals. One of the main contributions of the paper is the new concept of weak barriers, which only encodes orientation information of a barrier, disregarding the relative positions of its parts. We characterize weak barriers geometrically in terms of the convexification of $B$. Convex geometric tools then allow one to quantify the above mentioned stability for weak barriers in all dimensions.

math.MG↗
arXiv2025

Explicit lower bounds for opaque sets of unit square and unit disc

Explicit lower bounds for the length of the shortest opaque set for the unit disc and the unit square in the Euclidean plane are derived. The results are based on an explicit application of the general method of Kawamura, Moriyama, Otachi and Pach. Employing a recent observation by Steinerberger on the possible orientations of straight barriers with length close to Jones' bound, we improve the bound for the unit square by more than a factor $3$. The bound for barriers of the unit disc is new and based on the idea that the free parameters in the general method from can be optimized due to the strong symmetry properties of the disc. Our approach illustrates both the power and the limitations of the method.

math.MG↗
arXiv2025

Approximation of rearrangements by polarizations

The symmetric decreasing rearrangement of functions on $\mathbb{R}^n$ features in several seminal inequalities, such as the P\'olya-Szeg\H{o} inequality. The latter was shown by the authors to hold for all smoothing rearrangements, a class that includes the more general $(k,n)$-Steiner rearrangement, as well as others introduced by Brock and by Solynin. The theory of rearrangements and their associated set maps is developed, with an emphasis on approximation, particularly by polarizations. The P\'olya-Szeg\H{o} inequality holds with equality for polarizations, so is proved relatively easily for rearrangements that can be suitably approximated by them. One goal here is to show that the Brock rearrangements cannot be approximated in such a way. It turns out that under mild conditions, each set map associated with a rearrangement has in turn an associated contraction map from $\mathbb{R}^n$ to $\mathbb{R}^n$. With this new analytical tool, several general results on the approximation of rearrangements are also proved.

math.FA↗
arXiv2025

On the (outer) Minkowski content with lower-dimensional structuring element

Given a convex body $Q$ (structuring element) and a set $A$ in a Euclidean space, we consider the $Q$-Minkowski content of $A$. It is defined as the usual isotropic Minkowski content of $A$, but where the Euclidean ball is replaced by $Q$. When $Q$ is full-dimensional, the existence of the $Q$-Minkowski content can be assured by a sufficient condition which was stated by Ambrosio, Fusco and Pallara in the isotropic case. If $Q$ is not full-dimensional, we show that a weaker condition is sufficient for this purpose. We also consider the outer $Q$-Minkowski content of $A$ yielding the anisotropic perimeter of $A$ and we find a sufficient condition for its existence. Finally, we present an example of a set in three-dimensional Euclidean space, which does not admit the isotropic outer Minkwski content, but it admits the outer $Q$-Minkowski content for all two-dimensional disks $Q$.

math.MG↗
arXiv2024

A Blaschke-Petkantschin formula for linear and affine subspaces with application to intersection probabilities

Consider a uniformly distributed random linear subspace $L$ and a stochastically independent random affine subspace $E$ in $\mathbb{R}^n$, both of fixed dimension. For a natural class of distributions for $E$ we show that the intersection $L\cap E$ admits a density with respect to the invariant measure. This density depends only on the distance $d(o,E \cap L)$ of $L\cap E$ to the origin and is derived explicitly. It can be written as the product of a power of $d(o,E \cap L)$ and a part involving an incomplete beta integral. Choosing $E$ uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable $d(o,E \cap L)$ and study the behavior of the probability that $E \cap L$ hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where $E$ is tangent to the unit sphere, in which case we again derive the density for $d(o,E \cap L)$. Our probabilistic results are derived by means of a new integral-geometric transformation formula of Blaschke--Petkantschin type.

math.MG↗
arXiv2023

Rotational Crofton Formulae with a Fixed Subspace

The classical Crofton formula explains how intrinsic volumes of a convex body $K$ in $n$-dimensional Euclidean space can be obtained from integrating a measurement function at sections of $K$ with invariantly moved affine flats. Motivated by stereological applications, we present variants of Crofton's formula, where the flats are constrained to contain a fixed linear subspace $L_0$, but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case $\dim L_0=0$. The proof combines a suitable Blaschke--Petkantschin formula with the classical Crofton formula. We also argue that our main result is best possible, in the sense that one cannot estimate intrinsic volumes of a set, based on lower-dimensional sections, other than those given by our result. Finally, we provide a proof for a well-established variant: an integral relation for vertical sections. Our formula is stated for intrinsic volumes of a given set, complementing the classical approach for Hausdorff measures.

math.MG↗
arXiv2022

The P\'olya-Szeg\H{o} inequality for smoothing rearrangements

A basic version of the P\'olya-Szeg\H{o} inequality states that if $\Phi$ is a Young function, the $\Phi$-Dirichlet energy -- the integral of $\Phi(\|\nabla f\|)$ -- of a suitable function $f\in \mathcal{V}(\mathbb{R}^n)$, the class of nonnegative measurable functions on $\mathbb{R}^n$ that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions $f\in \mathcal{V}(\mathbb{R}^n)$, functions $f\in W^{1,p}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$ (when $1\le p<\infty$ and $\Phi(t)=t^p$), and functions $f\in W^{1,1}_{loc}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$. In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the P\'olya-Szeg\H{o} inequality previously available under a common and very general framework.

math.FA↗
arXiv2021

Discrepancy of stratified samples from partitions of the unit cube

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbfΩ=(Ω_1,\ldots,Ω_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected ${\mathcal{L}_p}$-discrepancy, $\mathbb{E} {\mathcal{L}_p}(\mathcal{P}_{\mathbfΩ})^p$, of a point set $\mathcal{P}_\mathbfΩ$ generated from any equivolume partition $\mathbfΩ$ is always strictly smaller than the expected ${\mathcal{L}_p}$-discrepancy of a set of $N$ uniform random samples for $p>1$. For fixed $N$ we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. We illustrate our results with explicit constructions for small $N$. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every $N$.

math.ST↗
arXiv2019

Rearrangement and polarization

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on $\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.

math.MG↗
arXiv2017

Dilation volumes of sets of finite perimeter

This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation $A\oplus tQ$ as $t$ converges to zero. Here $A$ and $Q$ are subsets of $n$-dimensional Euclidean space, $A$ has finite perimeter and $Q$ is finite. If $Q$ consists of two points only, $x$ and $x+u$, say, this derivative coincides up to sign with the directional derivative of the covariogram of $A$ in direction $u$. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of $A$. We extend this result to finite sets $Q$ and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at zero. The proofs are based on approximation of the characteristic function of $A$ by smooth functions of bounded variation and showing corresponding formulas for them.

math.PR↗
arXiv2017

Voronoi-based estimation of Minkowski tensors from finite point samples

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

math.MG↗
arXiv2014

Surface tensor estimation from linear sections

From Crofton's formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the n-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators. These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting.

math.PR↗
arXiv2010

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k^n) noisy covariogram measurements to construct a convex polytope P_k that approximates K or its reflection -K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P_k and K or -K tends to zero as k tends to infinity.

math.MG↗
arXiv2006

Convergence of algorithms for reconstructing convex bodies and directional measures

We investigate algorithms for reconstructing a convex body $K$ in $\mathbb {R}^n$ from noisy measurements of its support function or its brightness function in $k$ directions $u_1,...,u_k$. The key idea of these algorithms is to construct a convex polytope $P_k$ whose support function (or brightness function) best approximates the given measurements in the directions $u_1,...,u_k$ (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian. It is shown that this procedure is (strongly) consistent, meaning that, almost surely, $P_k$ tends to $K$ in the Hausdorff metric as $k\to\infty$. Here some mild assumptions on the sequence $(u_i)$ of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the $L_2$ metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball. Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in $k$ directions $u_1,...,u_k$. Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.

math.ST↗