Convex Bodies Research Articles

On Discrete LOG-Brunn--Minkowski Type Inequalities

The conjectured log-Brunn--Minkowski inequality says that the volume of centrally symmetric convex bodies $K,L\subset\mathbb{R}^n$ satisfies ${vol}\bigl((1-\lambda)\cdot K+_0\lambda\cdot L\bigr) \geq{vol}(K)^{1-\lambda}{vol}(L)^\lambda$, $\lambda\in(0,1)$, and is known to be true in the plane and for particular classes of symmetric convex bodies in $\mathbb{R}^n$. In this paper, we get some discrete log-Brunn--Minkowski type inequalities for the lattice point enumerator. Among others, we show that if $K,L\subset\mathbb{R}^n$ are unconditional convex bodies and $\lambda\in(0,1)$, then ${G}_n((1-\lambda)\cdot(K+C_n)+_0\lambda\cdot(L+C_n)+(-\frac{1}{2},\frac{1}{2})^n) \geq{G}_n(K)^{1-\lambda}{G}_n(L)^\lambda, $ where $C_n=[-1/2,1/2]^n$. Neither $C_n$ nor $(-1/2,1/2)^n$ can be removed. Furthermore, it implies the (volume) log-Brunn--Minkowski inequality for unconditional convex bodies. The corresponding results in the $L_p$ setting for $0<p<1$ are also obtained.

Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined by the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies.We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

A note on the concurrent normal conjecture

It is conjectured since long that for any convex body \(K \in \mathbb{R}^n\) there exists a point in the interior of K which belongs to at least 2n normals from different points on the boundary of K. The conjecture is known to be true for n = 2, 3, 4.Motivated by a recent preprint of Y. Martinez-Maure [4], we give a short proof of his result: for dimension \(n\geq 3\), under mild conditions, almost every normal through a boundary point to a smooth convex body \(K \in \mathbb{R}^n\) contains an intersection point of at least 6 normals from different points on the boundary of K.

Strong Convexity of Feasible Sets in Off-line and Online Optimization

It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there is renewed interest in this topic for both off-line and online problems. In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization: We first show the equivalence of two notions of curvature, namely, strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this shows that the Frank–Wolfe–type method of Wang and Abernethy has accelerated convergence rate [Formula: see text] over strongly convex feasible sets without additional assumptions on the (convex) objective function. In online linear optimization, we identify two main properties that help explaining why/when follow the leader (FTL) has only logarithmic regret over strongly convex sets. This allows one to directly recover and slightly extend a recent result of Huang et al., and to show that FTL has logarithmic regret over strongly convex sets whenever the gain vectors are nonnegative. We provide an efficient procedure for approximating convex bodies by strongly convex ones while smoothly trading off approximation error and curvature. This allows one to extend the improved algorithms over strongly convex sets to general convex sets. As a concrete application, we extend results on online linear optimization with hints to general convex sets. Funding: This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant Bolsa de Produtividade em Pesquisa #4310516/2017-0] Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Finance Code 001) Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (Jovem Cientista do Nosso Estado), Bolsa de Produtividade em Pesquisa #312751/2021-4 from CNPq, FAPERJ [Grant “Jovem Cientista do Nosso Estado”].

On the Lp Aleksandrov problem for negative p

Huang, Lutwak, Yang, and Zhang introduced the Lp integral curvature and posed the corresponding Lp Aleksandrov problem, the natural Lp extension of the classical integral curvature and Aleksandrov problem respectively. The problem asks about the existence of a convex body with prescribed Lp integral curvature measure. For the case of given even measures, the question will be solved for p∈(−1,0). Furthermore, a sufficient measure concentration condition will be provided for the case of p≤−1, again provided that the given measure is even.

Danzer's problem, effective constructions of dense forests and digital sequences

A 1965 problem due to Ludwig Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume 1. A recent approach to this problem is concerned with the construction of dense forests and is obtained by a suitable weakening of the volume constraint. A dense forest is a discrete point set of finite density getting uniformly close to long enough line segments. The distribution of points in a dense forest is then quantified in terms of a visibility function. Another way to weaken the assumptions in Danzer's problem is by relaxing the density constraint. In this respect, a new concept is introduced in this paper, namely that of an optical forest. An optical forest in R d $\mathbb {R}^{d}$ is a point set with optimal visibility but not necessarily with finite density. In the literature, the best constructions of Danzer sets and dense forests are not deterministic. The goal of this paper is to provide deterministic constructions of dense and optical forests which yield the best known results in any dimension d ⩾ 2 $d \geqslant 2$ in terms of visibility and density bounds, respectively. Namely, there are three main results in this work: (1) the construction of a dense forest with the best known visibility bound which, furthermore, enjoys the property of being deterministic; (2) the deterministic construction of an optical forest with a density failing to be finite only up to a logarithm and (3) the construction of a planar Peres-type forest (that is, a dense forest obtained from a construction due to Peres) with the best known visibility bound. This is achieved by constructing a deterministic digital sequence satisfying strong dispersion properties.

Polar duality between pairs of transversal Lagrangian planes; applications to uncertainty principles

We extend the notion of polar duality to pairs (ℓ,ℓ′) of transversal Lagrangian planes in the standard symplectic space. (R2n,ω). This allows us to show that polar duality has a natural symplectic interpretation. Our main results are the following: we first show that the oblique projections Ωℓ and Ωℓ′ on ℓ and ℓ′ of a centrally symmetric convex body Ω containing a symplectic ball with radius one satisfy the duality relation Ωℓo⊂Ωℓ′. We thereafter show that if, conversely, (Ωℓ,Ωℓ′)⊂ℓ×ℓ′ is such that Ωℓo=Ωℓ′ then Ωℓ and Ωℓ′ are the projections of a symplectic ball with radius one which is the John ellipsoid of Ωℓ×Ωℓ′o. We apply these results to the quantum principle of indeterminacy, which contains as a particular case the usual uncertainty principle.

Isobarycentric Inequalities

Abstract We prove the following isoperimetric-type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb {R}}^n$, half-spaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.

Size always matters, shape matters only for the big: potential optical effects of silica bodies in Selaginella.

Silica bodies are commonly found in Selaginella, but their function is unclear. Lens-like appearance and location in many species above giant chloroplasts of dorsal epidermal cells suggest optical functions. Silica body morphology in three Selaginella species was studied by microscopy. Optical effects were assessed by wave-optic simulations. Large convex, approximately hemispherical (papillose) and small approximately conical (concave–convex) silica bodies were found in different species. Both types lead to a concentrated spot of light high in the dorsal epidermal cell. Large convex bodies concentrate light 10–25 times in a shape-dependent manner by refraction, and small silica bodies concentrate light in a shape-insensitive, but wavelength-dependent, manner by diffraction (red light: approx. 2.3 times; blue light: approx. 1.5 times). Due to chloroplast movement, this concentrated light is above the chloroplast under high light, but within it under low light. Beyond the spot of concentration, light is dispersed into the chloroplast. Thin Selaginella leaves mean these effects may enhance light capture and minimize photodamage, but other effects such as inhibition of herbivory, mechanical support, and immune responses need to be considered. Silica bodies undoubtedly have optical effects, but their significance to the functioning of the plant requires direct studies of ecophysiological performance.

Cells in the box and a hyperplane

It is well known that a line can intersect at most 2n-1 cells of the n \times n chessboard. Here we consider the high-dimensional version: how many cells of the d -dimensional n\times \cdots \times n box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact: if K,L are convex bodies in \mathbb{R}^d and K\subset L , then the surface area of K is smaller than that of L .

Two new species of the subgenus Pleurosticha of the genus Chrysolina (Coleoptera: Chrysomelidae) from the mountains of southern Siberia (Altai and Tyva)

Two new species from the subgenus Pleurosticha of the genus Chrysolina are described from the mountains of southern Siberia (Russia): Ch. reshetnikovi sp. nov. (Republic of Altai) and Ch. borisi sp. nov. (Republic of Tyva). Chrysolina reshetnikovi sp. nov. has a slightly convex body, a blue-green coloration of the upper side, regular puncture rows of the elytra and mostly flat elytral intervals, which distinguish this species from all species of the subgenus, except for Ch. shapkini Mikhailov et Gus’kova, 2013. The main distinguishing features of the male of the new species from Ch. shapkini are the strongly broadened tarsomeres 1–3, the pygidium with an impression along its entire length, the last abdominal sternite with a semicircular depression, and the apex of the aedeagus bent at 90°. Chrysolina borisi sp. nov. has a convex small body, black coloration of the upper side, femora and tibiae, regular puncture rows of the elytra and slightly convex elytral intervals, and the aedeagus near the apex with a narrow protrusion of the lateral side directed toward the aedeagus base.

A New Species of the Genus Amberophytum Yu, Slipinski et Pang, 2019 (Coleoptera: Cerophytidae) from mid-Cretaceous Burmese Amber

This article describes and illustrates a new species of the genus Amberophytum Yu, Slipinski et Pang, 2019 of the family Cerophytidae from mid- Cretaceous Burmese Amber. The new species, A. maculatum s.n. differs from A. birmanicum Yu, Slipinski et Pang, 2019 in the smaller body size, shorter metatarsomere 1, and more convex body. A key to the species of the genus Amberophytum is presented.

Sections and projections of nested convex bodies

One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide characterizations of the sphere and the ellipsoid in terms of the properties of its sections or projections. Another kind of characterization of the ellipsoid is when we consider properties of the support cones. However, in almost all the known characterizations, we have only a convex body and the sections, projections, or support cones are considered for this given body. In this article we prove some results that characterize the Euclidean ball or the ellipsoid when the sections or projections are taken for a pair of nested convex bodies, i.e., two convex bodies K, L such that $$L\subset $$ int K. We impose some relations between the corresponding sections or projections and some apparently new characterizations of the ball or the ellipsoid appear. We also deal with properties of support cones or point source shadow boundaries when the apexes are taken in the boundary of K.

On the sine polarity and the Lp-sine Blaschke-Santaló inequality

This paper is dedicated to study the sine version of polar bodies and establish the Lp-sine Blaschke-Santaló inequality for the Lp-sine centroid body.The Lp-sine centroid body ΛpK for a star body K⊂Rn is a convex body based on the Lp-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of Λp∘K, the polar body of ΛpK, in terms of the volume of K. Thus, this inequality can be viewed as the “sine cousin” of the Lp Blaschke-Santaló inequality established by Lutwak and Zhang. As p→∞, the limit of Λp∘K becomes the sine polar body K⋄ and hence the Lp-sine Blaschke-Santaló inequality reduces to the sine Blaschke-Santaló inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies Cen, which consists of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in Cen are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santaló inequality are settled.

The Functional Form of Mahler’s Conjecture for Even Log-Concave Functions in Dimension 2

Abstract Let $\varphi :{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be an even convex function and ${\mathcal {L}}{\varphi }$ be its Legendre transform. We prove the functional form of Mahler’s conjecture concerning the functional volume product $P(\varphi )=\int e^{-\varphi }\int e^{-{\mathcal {L}}\varphi }$ in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case. The proof uses the computation of the derivative in $t$ of $P(t\varphi )$ and ideas due to Meyer [16] for unconditional convex bodies, adapted to the functional case by Fradelizi and Meyer [6] and extended for symmetric convex bodies in dimension 3 by Iriyeh and Shibata [11] (see also [4]).

Homothetic covering of convex hulls of compact convex sets

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $\lambda\in[0,1]$ such that $K$ can be covered by $m$ translates of $\lambda K$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.

Interpolating between volume and lattice point enumerator with successive minima

We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke’s shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski’s upper bound on the volume in terms of the successive minima.

Inverse Mean Curvature Flow With Singularities

Abstract This paper concerns the inverse mean curvature flow (IMCF) running from the boundary of a convex body that has no regularity assumption. We study the evolution of singularities by looking at the blow-up tangent cone around each singular point. We prove that the cone also evolves by the IMCF and that each singularity is removed when the evolving cone becomes flat. As a result, we derive the exact waiting time for a weak solution to become a smooth solution. In particular, necessary and sufficient condition for the existence of a smooth classical solution is given.

Covering Convex Bodies and the Closest Vector Problem

We present algorithms for the (1+epsilon )-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms in dimension n has running time of 2^{O(n)}(1/epsilon )^n. We improve this substantially in the following two cases. First, for ell _p-norms with p>2 (resp. p in [1,2]) fixed, we present an algorithm with a running time of 2^{O(n)}(1+1/epsilon )^{n/2} (resp. 2^{O(n)} (1+1/epsilon )^{n/p}). This result is based on a geometric covering problem, that was introduced in the context of CVP by Eisenbrand et al.: How many convex bodies are needed to cover the ball of the norm such that, if scaled by factor 2 around their centroids, each one is contained in the (1+epsilon )-scaled homothet of the norm ball? We provide upper bounds for this (2,epsilon )-covering number by exploiting the modulus of smoothness of the ell _p-balls. Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a (1+epsilon )-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net. Second, we consider polyhedral and zonotopal norms. For centrally symmetric polytopes (resp. zonotopes) in {mathbb R}^n with O(n) facets (resp. generated by O(n) line segments), we provide a deterministic O(log _2(2+1/epsilon ))^{O(n)} time algorithm. This generalizes the result of Eisenbrand et al. which applies to the ell _infty -norm. Finally, we establish a connection between the modulus of smoothness and lattice sparsification. As a consequence, using the enumeration and sparsification tools developped by Dadush, Kun, Peikert, and Vempala, we present a simple alternative to the boosting procedure with the same time and space requirement for ell _p norms. This connection might be of independent interest.

Simple Proof of the Lower Bound on the Average Distance from the Fermat-Weber Center of a Convex Body

Simple Proof of the Lower Bound on the Average Distance from the Fermat-Weber Center of a Convex Body