Intrinsic Volumes Research Articles

The Hadwiger Theorem on Convex Functions, I

AbstractA complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb{R}}^{n}$ is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

Stereologically adapted Crofton formulae for tensor valuations

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. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.

Iterated-logarithm laws for convex hulls of random walks with drift

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan [Probab. Theory Related Fields 93 (1992), pp. 377–392]. Our starting point is a version of Strassen’s functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.

Isothermal compressibility, internal pressure studies of aqueous rare earth nitrate solutions at 298.15 K: An attempt to estimate ionic radii and understand inner– and outer–sphere exchange water structural interactions

In this communication, we report our analysis and results concerning the determination of ionic radii in aqueous solution phase for Lanthanum (La+3), Neodymium (Nd+3), Erbium (Er+3) and Ytterbium (Yb+3) ions in presence of nitrate (NO3-) anion at 298.15 K. The literature data for density, adiabatic compressibility, and specific heat capacity of solutions at constant pressure have been used to compute isothermal compressibility, internal pressure of solutions in limiting concentration range. The apparent molar isothermal compressibility of electrolytes and hydration numbers have been obtained. The electrostriction for rare earth cations and intrinsic volumes have been calculated using thermodynamic equations. The ionic radii of cations have been obtained using the ionic partial molar volume data of NO3- ions and the results are compared with those obtained for rare earth chlorides. The two series affect is also observed for ionic radii and internal pressure. The results have been explained on the basis of water structural effects due to electrostatic field and water dipole interaction between the inner– and outer–sphere water exchange reaction.

Tessellation-valued processes that are generated by cell division

AbstractProcesses of random tessellations of the Euclidean space $\mathbb{R}^d$ , $d\geq 1$ , are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells.

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 L0, but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case dim⁡L0=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.

Angle Sums of Random Polytopes

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

Minkowski's successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

Random Zonotopes and Valuations

We define a random zonotope in $${\mathbb R}^d$$ , by adding finitely many random segments, which are independently and identically distributed. For this random polytope, we determine, under a mild assumption on the distribution, the expectations of the intrinsic volumes, more generally, the expectations of suitable valuations. We also prove a central limit theorem for a valuation evaluated at these random zonotopes.

Volumetric Properties of a Solution of tert-Butyl Alcohol in Carbon Tetrachloride: MD Modeling

Models of solutions of tert-butyl alcohol (TBA) in carbon tetrachloride (CTC) are obtained via all-atom molecular dynamics modeling. The excess volume of the solution and the apparent and intrinsic (geometric) volumes of both components are calculated throughout the range of concentrations. It is shown that the apparent and intrinsic molar volumes of TBA in the limit of low concentrations in solution are notably larger than in pure alcohol. At the same time, their values fall rapidly in a narrow range of concentrations (from 0 to 0.1 mole fractions), and then move almost linearly to their limit values in alcohol. It is found that such behavior of the volumetric characteristics of TBA is due to the specific association of alcohol at low concentrations because of the hydrogen bonding among TBA molecules.

Projections and Angle Sums of Belt Polytopes and Permutohedra

Let Psubset mathbb {R}^n be a belt polytope, that is a polytope whose normal fan coincides with the fan of some hyperplane arrangement {mathcal {A}}. Also, let G:mathbb {R}^nrightarrow mathbb {R}^d be a linear map of full rank whose kernel is in general position with respect to the faces of P. We derive a formula for the number of j-faces of the “projected” polytope GP in terms of the j-th level characteristic polynomial of {mathcal {A}}. In particular, we show that the face numbers of GP do not depend on the linear map G provided a general position assumption is satisfied. Furthermore, we derive formulas for the sum of the conic intrinsic volumes and Grassmann angles of the tangent cones of P at all of its j-faces. We apply these results to permutohedra of types A and B, which yields closed formulas for the face numbers of projected permutohedra and the generalized angle sums of permutohedra in terms of Stirling numbers of both kinds and their B-analogues.

Isoperimetric problems for zonotopes

AbstractShephard (Canad. J. Math. 26 (1974), 302–321) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note, we prove a generalization of this theorem yielding similar formulae for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for d‐dimensional zonotopes generated by d or segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the ℓ1 polarization problem on the unit sphere and to a vector‐valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.

Magnitude and Holmes–Thompson intrinsic volumes of convex bodies

AbstractMagnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.

New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Let \(\{X_i\}\) be a sequence of compact n-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov–Hausdorff sense to a compact Alexandrov space X. The paper (Alesker in Arnold Math J 4(1):1–17, 2018) outlined (without a proof) a construction of an integer-valued function on X; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of the \(X_i\). In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.

On p-Brunn–Minkowski inequalities for intrinsic volumes, with 0le p

We prove the validity of the p-Brunn–Minkowski inequality for the intrinsic volume V_k, k=2,dots , n-1, of symmetric convex bodies in {{mathbb {R}}}^n, in a neighbourhood of the unit ball when one of the bodies is the unit ball, for 0le p<1. We also prove that this inequality does not hold true on the entire class of convex bodies of {{mathbb {R}}}^n, when p is sufficiently close to 0.

Microstructure characterization and stochastic modeling of open‐cell foam based on μCT‐image analysis

AbstractFoam is a cellular material whose mechanical properties are strongly determined by its complex microstructure. To study the microstructure, at first a foam characterization based on CT image processing is required. Here we present an image segmentation procedure and determine the foam's characteristics using the lattice cell‐based concept of intrinsic volumes. Information like porosity, pore size distribution, and ligament shape are derived. These data are then employed as input for the generation of stochastic foam volume elements with the corresponding morphology. The introduced microstructural characterization and foam generation procedures are validated by an inverse analysis, that is, by a CT image analysis of the stochastic foam volume element. Additionally, an example investigation of industrial polyurethane foam proves the concepts.

Contextual Search via Intrinsic Volumes

We study the problem of contextual search, a multidimensional generalization of binary search that captures many problems in contextual decision-making. In contextual search, a learner is trying to learn the value of a hidden vector $v \in [0,1]^d$. Every round the learner is provided an adversarially-chosen context $u_t \in \mathbb{R}^d$, submits a guess $p_t$ for the value of $\langle u_t, v\rangle$, learns whether $p_t < \langle u_t, v\rangle$, and incurs loss $\ell(\langle u_t, v\rangle, p_t)$ (for some loss function $\ell$). The learner's goal is to minimize their total loss over the course of $T$ rounds. We present an algorithm for the contextual search problem for the symmetric loss function $\ell(\theta, p) = |\theta - p|$ that achieves $O_{d}(1)$ total loss. We present a new algorithm for the dynamic pricing problem (which can be realized as a special case of the contextual search problem) that achieves $O_{d}(\log \log T)$ total loss, improving on the previous best known upper bounds of $O_{d}(\log T)$ and matching the known lower bounds (up to a polynomial dependence on $d$). Both algorithms make significant use of ideas from the field of integral geometry, most notably the notion of intrinsic volumes of a convex set. To the best of our knowledge this is the first application of intrinsic volumes to algorithm design.

The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge–Ampère measures

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge–Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.

Convex cones spanned by regular polytopes

Abstract We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

Beta-star polytopes and hyperbolic stochastic geometry

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in Rd with density proportional to (‖x‖2−1)−β, where ‖x‖>1 and β>d/2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f-vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry.