arXiv subjects

Daniel Hug

Publications and source records attributed to Daniel Hug.

At least 19 recordsLinked to original sources

arXiv2026

Lengths and incidences in Poisson hypersphere and spherical splitting tessellations

We investigate distributional properties of two random tessellation models on the $d$-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For $d=2$, we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.

math.PR↗

arXiv2025

High-dimensional limits arising from hyperbolic Poisson k-plane processes

We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space $\mathbb H^d$ of constant curvature $-1$, with $d \ge 4$ and $1 \le k \le d-1$. It is known that, after centring and normalization, the total $k$-volume of all intersections of $k$-planes with a geodesic ball of radius $R$ converges in distribution, as $R \to \infty$, to a non-Gaussian infinitely divisible random variable $Z_{d,k}$ whenever $2k > d+1$. We investigate the distributional behaviour of $Z_{d,k}$ in the high-dimensional regime $d \to \infty$ and depending on how fast $k$ grows in relation to $d$. We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the L\'evy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension $d-k$ and a standard Gaussian limit for $d-k \to \infty$.

math.PR↗

arXiv2025

Minkowski tensors for point clouds and voxelized data: robust, asymptotically unbiased estimators

Minkowski tensors, also known as tensor valuations, provide robust $n$-point information for a wide range of random spatial structures. Local estimators for point clouds, e.g., representing voxelized data, however, are unavoidably biased even in the limit of infinitely high resolution. Here, we substantially improve a recently proposed, asymptotically unbiased algorithm to estimate Minkowski tensors from point clouds. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors, among others, to the case of finite unions of compact sets with positive reach, which is relevant for many applications like rough surfaces or composite materials. As a realistic test case of random spatial structures, we consider random (beta) polytopes. We first derive explicit expressions of the expected Minkowski tensors, which we then compare to our simulation results. We obtain precise estimates with relative errors of a few percent for practically relevant resolutions. Finally, we apply our methods to real data of metallic grains and nanorough surfaces, and we provide an open-source python package, which works in any dimension.

math.ST↗

arXiv2025

Visibility and intersection density for Boolean models in hyperbolic space

For Poisson particle processes in hyperbolic space we introduce and study concepts analogous to the intersection density and the mean visible volume, which were originally considered in the analysis of Boolean models in Euclidean space. In particular, we determine a necessary and sufficient condition for the finiteness of the mean visible volume of a Boolean model in terms of the intensity and the mean surface area of the typical grain.

math.PR↗

arXiv2024

Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies

Barthe, Schechtman and Schmuckenschl\"ager proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove close to be optimal stronger stability versions of these results, together with their counterparts about the $\ell$-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of even isotropic measures on the unit sphere.

math.MG↗

arXiv2024

Intersections of Poisson k-flats in hyperbolic space: completing the picture

In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of $k$-flats in $d$-dimensional hyperbolic space $\mathbb{H}^d$, for $0\le k\le d-1$. A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total $k$-dimensional volume $F_r$ of the process inside a spherical observation window $B_r$ of radius $r$ when one lets $r$ tend to infinity. While $F_r$ is asymptotically normally distributed for $2k\leq d+1$, it has been shown to obey a nonstandard central limit theorem for $2k>d+1$. The intersection process of order $m$, for $d-m(d-k) \geq 0$, of the original process $\eta$ consists of all intersections of distinct flats $E_1,\ldots,E_m \in \eta$ with $\dim(E_1\cap\ldots\cap E_m) = d-m(d-k)$. For this intersection process, the total $d-m(d-k)$-dimensional volume $F^{(m)}_r$ of the process in $B_r$, again as $r \to \infty$, is of particular interest. For $2k \leq d+1$ it has been shown that $F^{(m)}_r$ is again asymptotically normally distributed. For $m \geq 2$, the limit is so far unknown, although it has been shown for certain $d$ and $k$ that it cannot be a normal distribution. We determine the limit distribution for all values of $d,k,m$. In addition, we establish explicit rates of convergence in the Kolmogorov distance and discuss properties of the limit distribution. Furthermore we show that the asymptotic covariance matrix of the vector $(F^{(1)}_r,\ldots,F^{(m)}_r)^\top$ has full rank when $2k < d+1$ and rank one when $2k \geq d+1$.

math.PR↗

arXiv2024

Boolean models in hyperbolic space

The union of the particles of a stationary Poisson process of compact (convex) sets in Euclidean space is called Boolean model and is a classical topic of stochastic geometry. In this paper, Boolean models in hyperbolic space are considered, where one takes the union of the particles of a stationary Poisson process in the space of compact (convex) subsets of the hyperbolic space. Geometric functionals such as the volume of the intersection of the Boolean model with a compact convex observation window are studied. In particular, the asymptotic behavior for balls with increasing radii as observation windows is investigated. Exact and asymptotic formulas for expectations, variances, and covariances are shown and univariate and multivariate central limit theorems are derived. Compared to the Euclidean framework, some new phenomena can be observed.

math.PR↗

arXiv2024

Additive kinematic formulas for convex functions

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Amp\`ere measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Amp\`ere measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered.

math.MG↗

arXiv2024

Kubota-type formulas and supports of mixed measures

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Amp\`ere measures. As a by-product, we characterize supports of mixed area and mixed Monge-Amp\`ere measures, thereby confirming a special case of a conjecture by Schneider.

math.MG↗

arXiv2023

Extremizers of the Alexandrov--Fenchel inequality within a new class of convex bodies

Mixed volumes in $n$-dimensional Euclidean space are functionals of $n$-tuples consisting of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The Alexandrov--Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies, which cover as very special cases many important inequalities between basic geometric functionals. The problem of characterizing completely the equality cases in the Alexandrov--Fenchel inequality is wide open. Major recent progress was made by Yair Shenfeld and Ramon van Handel \cite{SvH22,SvH23+}, in particular they resolved the problem in the cases where $K,L$ are general convex bodies and $C_1,\ldots,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some dimensional restriction). We introduce the class of polyoids, which includes polytopes, zonoids and triangle bodies, and characterize polyoids by using generating measures. Based on this characterization and Shenfeld and van Handel's contribution, we extend their result to a class of convex bodies containing all polyoids and smooth bodies. Our result is stated in terms of the support of the mixed area measure of the unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. A geometric description of this support is provided in the accompanying work \cite{HugReichert23+}.

math.MG↗

arXiv2023

The support of mixed area measures involving a new class of convex bodies

Mixed volumes in $n$-dimensional Euclidean space are functionals of $n$-tuples of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The Alexandrov--Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov--Fenchel inequality remains a challenging open problem. Major recent progress was made by Yair Shenfeld and Ramon van Handel \cite{SvH22,SvH23+}, in particular they resolved the problem in the cases where $C_1,\ldots,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some dimensional restriction). In \cite{HugReichert23+} we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution, we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary $(n-1)$-tuple of polyoids (or smooth bodies). The result confirms a long-standing conjecture by Rolf Schneider in the case of polyoids and hence, in particular, of zonoids.

math.MG↗

arXiv2023

Boolean models

The topic of this survey are geometric functionals of a Boolean model (in Euclidean space) governed by a stationary Poisson process of convex grains. The Boolean model is a fundamental benchmark of stochastic geometry and continuum percolation. Moreover, it is often used to model amorphous connected structures in physics, materials science and biology. Deeper insight into the geometric and probabilistic properties of Boolean models and the dependence on the underlying Poisson process can be gained by considering various geometric functionals of Boolean models. Important examples are the intrinsic volumes and Minkowski tensors. We survey here local and asymptotic density (mean value) formulas as well as second order properties and central limit theorems.

math.PR↗

arXiv2023

Vectorial analogues of Cauchy's surface area formula

Cauchy's surface area formula says that for a convex body $K$ in $n$-dimensional Euclidean space the mean value of the $(n-1)$-dimensional volumes of the orthogonal projections of $K$ to hyperplanes is a constant multiple of the surface area of $K$. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation on convex bodies.

math.MG↗

arXiv2023

Intersections of Poisson $ k $-flats in constant curvature spaces

Poisson processes in the space of $k$-dimensional totally geodesic subspaces ($k$-flats) in a $d$-dimensional standard space of constant curvature $\kappa\in\{-1,0,1\}$ are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order $m$ together with their $(d-m(d-k))$-dimensional Hausdorff measure within a geodesic ball of radius $r$. Asymptotic normality for fixed $r$ is shown as the intensity of the underlying Poisson process tends to infinity for all $m$ satisfying $d-m(d-k)\geq 0$. For $\kappa\in\{-1,0\}$ the problem is also approached in the set-up where the intensity is fixed and $r$ tends to infinity. Again, if $2k\le d+1$ a central limit theorem is shown for all possible values of $m$. However, while for $\kappa=0$ asymptotic normality still holds if $2k>d+1$, we prove for $\kappa=-1$ convergence to a non-Gaussian infinitely divisible limit distribution in the special case $m=1$. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin--Stein method. We also show for general $\kappa\in\{-1,0,1\}$ that, roughly speaking, the variances within a general observation window $W$ are maximal if and only if $W$ is a geodesic ball having the same volume as $W$. Along the way we derive a new integral-geometric formula of Blaschke--Petkantschin type in a standard space of constant curvature.

math.PR↗

arXiv2022

Curvature measures and soap bubbles beyond convexity

Extending the celebrated results of Alexandrov (1958) and Korevaar-Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on the natural geometric condition involving the curvature measures. More specifically we show, under a natural mean-convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach $ A \subseteq \mathbf{R}^{n+1} $ with finite and positive volume such that, for some $ k \in \{0, \ldots , n-1\}$, the $ k $-th generalized curvature measure $ \Theta^\phi_{k}(A, \cdot) $, which is defined on the unit normal bundle of $ A $ with respect to the relative geometry induced by $ \phi $, is proportional to $ \Theta^\phi_{n}(A, \cdot)$. If $ k = n-1 $ the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach. These results are new even in the Euclidean space. Several auxiliary and related results are also proved, which are of independent interest. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space and a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle--Lussardi--Villa (2021).

math.MG↗

arXiv2021

On a game of chance in Marc Elsberg's thriller "GREED"

A (possibly illegal) game of chance, which is described in Chapter 14 of Marc Elsberg's thriller "GREED", seems to offer an excellent chance of winning. However, as the gambling starts and evolves over several rounds, the actual experience of the vast majority of the gamblers in a pub is strikingly different. We provide an analysis of this specific game and several of its variants by elementary tools of probability. Thus we also encounter an interesting threshold phenomenon, which is related to the transition from a profit zone to a loss area. Our arguments are motivated and illustrated by numerical calculations with Python.

math.PR↗

arXiv2021

Reverse Alexandrov--Fenchel inequalities for zonoids

The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,\ldots,K_n)$ of convex bodies $K_1,\ldots,K_n$ in $\mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,\ldots,K_n)$ and $V(K_2,K_2,K_3,\ldots,K_n)$. As a consequence, for integers $\alpha_1,\ldots,\alpha_m\in\mathbb{N}$ with $\alpha_1+\cdots+\alpha_m=n$ the product $V_n(K_1)^{\frac{\alpha_1}{n}}\cdots V_n(K_m)^{\frac{\alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,\ldots,m$, is a lower bound for the mixed volume $V(K_1[\alpha_1],\ldots,K_m[\alpha_m])$, where $\alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[\alpha_1],\ldots,K_m[\alpha_m])$ in terms of the product of the intrinsic volumes $V_{\alpha_i}(K_i)$, for $i=1,\ldots,m$. The case where $m=2$, $\alpha_1=1$, $\alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $\alpha_1=\alpha_2=1$, $\alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $\alpha_2=\cdots=\alpha_m=1$, $K_1$ is the unit ball and $K_2,\ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.

math.MG↗

arXiv2021

Extremizers and stability of the Betke--Weil inequality

Let $K$ be a compact convex domain in the Euclidean plane. The mixed area $A(K,-K)$ of $K$ and $-K$ can be bounded from above by $1/(6\sqrt{3})L(K)^2$, where $L(K)$ is the perimeter of $K$. This was proved by Ulrich Betke and Wolfgang Weil (1991). They also showed that if $K$ is a polygon, then equality holds if and only if $K$ is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality $6\sqrt{3}A(K,-K)\le L(K)^2$.

math.MG↗