Intrinsic Volumes Research Articles

Convex hulls of multidimensional random walks

Let $S_k$ be a random walk in $\mathbb R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $\operatorname {conv} (S_1, \dots , S_n)$ of the first $n$ steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, $p_n$ does not depend on the distribution of increments. This extends the well-known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of $p_n$ as $n \to \infty$ for any planar random walk with zero mean square-integrable increments. We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension $d \ge 2$. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer–Widom formula (1961) on the perimeter of planar walks: \begin{equation*} \mathbb E V_1 (\operatorname {conv} (0, S_1, \dots , S_n)) = \sum _{k=1}^n \frac {\mathbb E \|S_k\|}{k}, \end{equation*} where $V_1$ denotes the first intrinsic volume, which is proportional to the mean width. These results have applications to geometry and, in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities $p_n$. We also prove similar results for convex hulls of random walk bridges and, more generally, for partial sums of exchangeable random vectors.

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of $$\alpha $$ -mixing (for local statistics) and exponential $$\alpha $$ -mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

For any closed smooth Riemannian manifold Weyl (Am J Math 61:461–472, 1939) has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called weakly smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures. The work is a joint project with Petrunin.

The Kneser–Poulsen Conjecture for Special Contractions

The Kneser–Poulsen conjecture states that if the centers of a family of N unit balls in $${\mathbb E}^d$$ are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $$N\ge (1+\sqrt{2})^d$$ . Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.

Random polytopes: Central limit theorems for intrinsic volumes

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein’s method from probability theory.

Effect of Impairment-Based Rehabilitation on Lower Leg Muscle Volumes and Strength in Patients With Chronic Ankle Instability: A Preliminary Study.

Context: Patients with chronic ankle instability (CAI) have demonstrated atrophy of foot and ankle musculature and deficits in ankle strength. The effect of rehabilitation on muscle morphology and ankle strength has not previously been investigated in patients with CAI. Objective: Our objective was to analyze the effect of impairment-based rehabilitation on intrinsic and extrinsic foot and ankle muscle volumes and strength in patients with CAI. Design: Controlled laboratory study. Setting: Laboratory. Patients: Five young adults with CAI. Intervention: Twelve sessions of supervised impairment-based rehabilitation that included range of motion, strength, balance, and functional exercises. Main Outcome Measures: Measures of extrinsic and intrinsic foot muscle volume and ankle strength measured before and after 4 weeks of supervised rehabilitation. Novel fast-acquisition magnetic resonance imaging was used to scan from above the femoral condyles through the entire foot. The perimeter of each muscle was outlined on each axial slice and then the 2-dimensional area was multiplied by the slice thickness (5mm) to calculate muscle volume. Plantar flexion, dorsiflexion, inversion, and eversion isometric strength were measured using a hand-held dynamometer. Results: Rehabilitation resulted in hypertrophy of all extrinsic foot muscles except for the flexor hallucis longus and peroneals. Large improvements were seen in inversion, eversion, and plantar flexion strength following rehabilitation. Effect sizes for significant differences following rehabilitation were all large and ranged from 1.54 to 3.35. No significant differences were identified for intrinsic foot muscle volumes. Conclusion: Preliminary results suggest that impairment-based rehabilitation for CAI can induce hypertrophy of extrinsic foot and ankle musculature with corresponding increases in ankle strength.

Intrinsic Entropies of Log-Concave Distributions

The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable $X$ , the sequence of the $\lfloor n\theta \rfloor ^{\text {th}}$ intrinsic volumes of the typical sets of $X$ in dimensions $n \geq 1$ grows exponentially with a well-defined rate. We denote this rate by $h_{X}(\theta )$ , and call it the $\theta ^{\text {th}}$ intrinsic entropy of $X$ . We show that $h_{X}(\theta )$ is a continuous function of $\theta $ over the range [0, 1], thereby providing a smooth interpolation between the values 0 and $h(X)$ at the endpoints 0 and 1, respectively.

Log-concavity properties of Minkowski valuations

New Orlicz Brunn–Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new classification of generalized translation invariant valuations.

Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss–Bonnet relations, and the principal kinematic formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.

Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

We give an explicit formula for the probability that the convex hull of an n-step random walk in {mathbb{R}^d} does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments.This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of {mathbb{R}^n} of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type {A_{n-1}} (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form {B_1times cdots times B_1} recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as {n to infty}, in both cases of fixed and increasing dimension d.

Detecting precursors of localization by strain-field analysis

Localization in the deformation field, even though initiated locally at the microscopic scale, leads upon increased deformation to fracture at the macroscopic scale, thereby violating the separation of length scales. Localization and damage can be accounted for in macroscopic modeling by appropriate enrichments at that level, however doing so requires (i) detecting the onset of localization prior to its actual occurrence and (ii) quantifying the kinematical characteristics of the localization band. This paper serves that goal. A methodology is developed to analyze the evolution of strain- and displacement-fields during deformation. A key ingredient in this analysis is the use of the Minkowski functionals (also known as intrinsic volumes, quermass integrals, or curvature integrals) from integral geometry, to detect emerging patterns in thresholded strain- and displacement-fields. Doing so, the onset of localization in the microstructure is detected as the emergence of a correlated and narrow pattern of high strains, prior to the actual loss of material stability. Furthermore, the developed localization band is characterized in terms of a weak displacement discontinuity, incorporating the width and direction of the band. The developed methodology uses kinematical fields only, and is therefore applicable to both numerical and experimental deformation-field data. For illustration purposes, numerical data from a finite-element simulation of a deformed voided microstructure is used, without any loss of generality.

Inclusion–Exclusion Principles for Convex Hulls and the Euler Relation

Consider n points \(X_1,\ldots ,X_n\) in \(\mathbb {R}^d\) and denote their convex hull by \({\Pi }\). We prove a number of inclusion–exclusion identities for the system of convex hulls \({\Pi }_I:=\mathrm{conv}(X_i: i\in I)\), where I ranges over all subsets of \(\{1,\ldots ,n\}\). For instance, denoting by \(c_k(X)\) the number of k-element subcollections of \((X_1,\ldots ,X_n)\) whose convex hull contains a point \(X\in \mathbb {R}^d\), we prove that $$\begin{aligned} c_1(X)-c_2(X)+c_3(X)-\cdots + (-1)^{n-1} c_n(X) = (-1)^{\dim {\Pi }} \end{aligned}$$ for allX in the relative interior of \({\Pi }\). This confirms a conjecture of Cowan (Adv Appl Probab 39(3):630–644, 2007) who proved the above formula for almost allX. We establish similar results for the number of polytopes \({\Pi }_J\) containing a given polytope \({\Pi }_I\) as an r-dimensional face, thus proving another conjecture of Cowan (Discrete Comput Geom 43(2):209–220, 2010). As a consequence, we derive inclusion–exclusion identities for the intrinsic volumes and the face numbers of the polytopes \({\Pi }_I\). The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler–Schlafli–Poincare relation and is of independent interest.

Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset\mathbb{R}^{d}$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,\ldots,d\}$, customarily denoted by $\mathcal{L}(V_{C})$. The aim of the present paper is to provide a Berry–Esseen bound for the normal approximation of $\mathcal{L}(V_{C})$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_{n}})$, $n\geq1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein’s method and second-order Poincare inequality, (3) concentration estimates and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp [Inf. Inference 3 (2014) 224–294] and McCoy and Tropp [Discrete Comput. Geom. 51 (2014) 926–963] about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.

Multiparametric Characterization of Single, Unlabeled Proteins in Solution

With large diversity in structure and function, as well as clinical relevance, proteins represent an important target for biophysical characterization. Resistive pulse-based nanopore sensing is a compelling platform for this task, as it provides information about individual proteins during their translocation through the zeptoliter sensing volume inside of a nanopore. Previous work in our group used lipid-coated synthetic nanopores to extract five distinct protein descriptors by analyzing modulations in ionic current, ΔI, resulting from the translocation and rotation of individual lipid-tethered proteins [1]. However, while coating of nanopore walls with fluid lipid bilayers and tethering proteins with lipid anchors is useful for preventing non-specific protein adsorption, extending translocation times, and increasing specificity, it is technically demanding and its success depends on nanopore diameter, geometry, and surface chemistry. This work explores the possibility of using the detergent Tween-20 for nanopore surface coating [2] combined with high bandwidth recording electronics to characterize freely translocating, untethered proteins on a single molecule level. Here, we utilize the dependence of ΔI on the orientation of non-spherical proteins transiting a nanopore to determine their intrinsic shapes, volumes, and dipole moments in solution. The ability to thoroughly examine unlabeled, natively-folded proteins in an aqueous sample on a single molecule level signifies an important step toward the use of nanopores for proteomic and diagnostic applications.[1] Yusko, E.C., et al., Real-time shape approximation and 5-D fingerprinting of single proteins. arXiv preprint arXiv:1510.01935, 2015.[2] Hu, Rui, et al. Intrinsic and membrane-facilitated α-synuclein oligomerization revealed by label-free detection through solid-state nanopores. Scientific reports 6, 2016.

Compressed sensing of data with a known distribution

Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted ℓ1-norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.

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 approximation of 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.

Intrinsic Volumes and Gaussian Polytopes: The Missing Piece of the Jigsaw

The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this missing piece of the jigsaw in the theory of Gaussian polytopes are discussed.

TMAO and urea in the hydration shell of the protein SNase.

We performed all-atom MD simulations of the protein SNase in aqueous solution and in the presence of two major osmolytes, trimethylamine-N-oxide (TMAO) and urea, as cosolvents at various concentrations and compositions and at different pressures and temperatures. The distributions of the cosolvent molecules and their orientation in the surroundings of the protein were analyzed in great detail. The distribution of urea is largely conserved near the protein. It varies little with pressure and temperature, and does practically not depend on the addition of TMAO. The slight decrease with temperature of the number of urea molecules that are in contact with the SNase molecule is consistent with the view that the interaction of the protein with urea is mainly of enthalpic nature. Most of the TMAO molecules tend to be oriented to the protein by its methyl groups, a small amount of these molecules contact the protein by its oxygen, forming hydrogen bonds with the protein, only. Unlike urea, the fraction of TMAO in the hydration shell of SNase slightly increases with temperature (a signature of a prevailing hydrophobic interaction between TMAO and SNase), and decreases significantly upon the addition of urea. This behavior reflects the diverse nature of the interaction of the two osmolytes with the protein. Using the Voronoi volume of the atoms of the solvent molecules (water, urea, TMAO), we compared the fraction of the volume occupied by a given type of solvent molecule in the hydration shell and in the bulk solvent. The volume fraction of urea in the hydration shell is more than two times larger than in the bulk, whereas the volume fraction of TMAO in the hydration shell is only slightly larger in the binary solvent (TMAO + water) and becomes even less than in the bulk in the ternary solvent (TMAO + water + urea). Thus, TMAO tends to be excluded from the hydration shell of the protein. The behavior of the two cosolvents in the vicinity of the protein does not change much with pressure (from 1 to 5000 bar) and temperature (from 280 to 330 K). This is also in line with the conception of the "osmophobic effect" of TMAO to protect proteins from denaturation also at harsh environmental conditions. We also calculated the volumetric parameters of SNase and found that the cosolvents have a small but significant effect on the apparent volume and its contributions, i.e. the intrinsic, molecular and thermal volumes.

Multivariate central limit theorems for Rademacher functionals with applications

Quantitative multivariate central limit theorems for general functionals of independent, possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application we prove a quantitative multivariate central limit theorem for vectors of intrinsic volumes induced by random cubical complexes.

Local Digital Algorithms Applied to Boolean Models

AbstractWe investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of local configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For three‐dimensional stationary isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains, we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.