Convex Bodies Research Articles

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

AbstractWe construct a differentiable locally Lipschitz function in with the property that for every convex body there exists such that coincides with the set of limits of derivatives of sequences converging to . The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite‐dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.

The curvature entropy inequalities of convex bodies

Abstract There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

Higher-order Lp isoperimetric and Sobolev inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

On the monotonicity of non-local perimeter of convex bodies

On the monotonicity of non-local perimeter of convex bodies

On the blocking numbers of some special convex bodies

On the blocking numbers of some special convex bodies

Lattice reduced and complete convex bodies

AbstractThe purpose of this paper is to study convex bodies for which there exists no convex body of the same lattice width. Such bodies will be called ‘lattice reduced’, and they occur naturally in the study of the flatness constant in integer programming, as well as other problems related to lattice width. We show that any simplex that realizes the flatness constant must be lattice reduced and prove structural properties of general lattice reduced convex bodies: they are polytopes with at most vertices and their lattice width is attained by at least independent directions. Strongly related to lattice reduced bodies are the ‘lattice complete bodies’, which are convex bodies for which there exists no such that has the same lattice diameter as . Similar structural results are obtained for lattice complete bodies. Moreover, various construction methods for lattice reduced and complete convex bodies are presented.

Tangential contact modeling to seal considering elastic-plastic for lubrication transition mechanism

Researches on the reciprocating seals lubrication under elastic-hydrodynamic lubrication (EHL) are mature, and leakage of failure criterion has been described with Couette flow for more than 30 years, which could not reflect to leakage mechanism in multiscale. Thus, it is necessary to explore seal leakage failure to focus on micro convex bodies of asperities model under contact pressure and cycle tangential friction stress in mixed lubrication. This is because convex bodies behaviors determine sealing effect and lubrication behaviors in microscale with subjecting to the dominant pressure. Interaction between convex bodies fatigue hysteresis curve and elastic-plastic strain are also calculated to explore sealing effect with leakage rate mechanism. We then design water seal at high pressure of 100 MPa with the speed of 0.1–1 m/s and medium temperature from 5 ℃ to 40 ℃. IWAN model of the convex bodies are calculated. Probability density function and Jenkin’s elements critical slip displacement are obtained. Convex bodies of radial and tangential deformation are also calculated through loading of contact pressure and friction stress with cycle strength coefficient to obtain seal fatigue lifespan. Ratio to seal stick and slip friction is calculated for evaluating the loading boundary. Seal leakage failure mechanism with fatigue characteristics is illustrated with macro lubrication parameters and micro convex bodies mechanics performance.

Nested barycentric coordinate system as an explicit feature map for polyhedra approximation and learning tasks

We introduce a new embedding technique based on a nested barycentric coordinate system. We show that our embedding can be used to transform the problems of polyhedron approximation, piecewise linear classification and convex regression into one of finding a linear classifier or regressor in a higher dimensional (but nevertheless quite sparse) representation. Our embedding maps a piecewise linear function into an everywhere-linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We explain the applications of our embedding to the problems of approximating separating polyhedra—in fact, it can approximate any convex body and unions of convex bodies—as well as to classification by separating polyhedra, and to piecewise linear regression.

Observation routes and external watchman routes

We introduce the Observation Route Problem (ORP) defined as follows: Given a set of n pairwise disjoint obstacles (regions) in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) each obstacle from some point of the tour. The observer does not need to see the entire boundary of an obstacle. The tour is not allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods (TSPN) and the External Watchman Route Problem (EWRP). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results:(I) Given a family of n disjoint convex bodies in the plane, computing a shortest observation route does not admit a (clog⁡n)-approximation unless P=NP for an absolute constant c>0. (This holds for both limited and unlimited vision.)(II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is NP-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.)(III) Given a family of n disjoint fat convex polygons in the plane, an observation tour whose length is at most O(log⁡n) times the optimal can be computed in polynomial time. (This holds for limited vision.)(IV) For every n≥5, there exists a convex polygon with n sides and all angles obtuse such that its perimeter is not a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).

Small perturbations of polytopes

Motivated by first-order conditions for extremal bodies of geometric functionals, we study a functional analytic notion of infinitesimal perturbations of convex bodies and give a full characterization of the set of realizable perturbations if the perturbed body is a polytope. As an application, we derive a necessary condition for polytopal maximizers of the isotropic constant.

Three-chromatic geometric hypergraphs

We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored in such a way that no translate of C contains m points of P (or more), all of the same color. As a part of the proof, we show a strengthening of the Erdős–Sands–Sauer–Woodrow conjecture. Surprisingly, the proof also relies on the two-dimensional case of the Illumination Conjecture. The extended abstract of this paper already appeared in the proceedings of SoCG ’22.

An open-source MATLAB toolbox for 3D block cutting and 3D mesh cutting in geotechnical engineering

This study presents a simple approach and its associated MATLAB toolbox for 3D block cutting and mesh cutting. The approach is suitable for the meshes of numerical methods including the Key Block Theory (KBT), the Discontinuous Deformation Analysis (DDA), the Numerical Manifold Method (NMM), and the cut Finite Element Method (Cut-FEM). The strategy is based on calculations on convex bodies. It uses two different forms of representation: the geometric representation which includes vertices and faces, and the algebraic representation which consists of inequalities. The cutting was implemented on the algebraic representation, and the resulting inequalities were converted into a geometric representation. The above strategy turned out to be robust and straightforward to execute, at the cost of a general body being regarded as a combination of convex bodies. The efficiency guarantee was considered through pre-checking algorithms. The source code was provided, as well as some simple examples.

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.

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,C1,…,Cn−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 [13,14], in particular they resolved the problem in the cases where C1,…,Cn−2 are polytopes, zonoids or smooth bodies (under some dimensional restriction). In [6] 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 (and under a dimensional restriction), 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 Bn and C1,…,Cn−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 thus (partially) confirms a long-standing conjecture by Rolf Schneider in the case of polyoids, and hence in particular it covers the case of zonoids and triangle bodies.

The general minimal Orlicz surface area for measures

Petty proved that a convex body in [Formula: see text] has the minimal surface area among its [Formula: see text] images if and only if its surface area measure is isotropic. In this paper, we seek the general minimal Orlicz surface area of a convex body containing the origin in its interior among its [Formula: see text] images for a measure [Formula: see text] on [Formula: see text] with density that has a positive degree of homogeneous, demonstrate the existence of the extreme body, and establish a necessary condition for the extreme body. Moreover, we also provide a counterexample to negate the uniqueness of the extreme body attaining the general minimal Orlicz surface area for [Formula: see text].

Some new minimax inequalities for centered convex bodies

AbstractThe purpose of this manuscript is to derive some new minimax inequalities for centered convex bodies in $${\mathbb {R}}^{d}$$ R d . As consequences, new characterizations of ellipsoids will be established as well. Some open problems related to the Busemann-Petty list of problems will also be discussed.

Reduced polygons in the hyperbolic plane

For a hyperplane H supporting a convex body C in the hyperbolic space Hd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^d$$\\end{document}, we define the width of C determined by H as the distance between H and a most distant ultraparallel hyperplane supporting C. The minimum width of C over all supporting H is called the thickness Δ(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (C)$$\\end{document} of C. A convex body R⊂Hd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R \\subset \\mathbb {H}^{d}$$\\end{document} is said to be reduced if Δ(Z)<Δ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (Z) < \\Delta (R)$$\\end{document} for every convex body Z properly contained in R. We describe a class of reduced polygons in H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2}$$\\end{document} and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.

On the variance of the mean width of random polytopes circumscribed around a convex body

AbstractLet be a convex body in in which a ball rolls freely and which slides freely in a ball. Let be the intersection of i.i.d. random half‐spaces containing chosen according to a certain prescribed probability distribution. We prove an asymptotic upper bound on the variance of the mean width of as . We achieve this result by first proving an asymptotic upper bound on the variance of the weighted volume of random polytopes generated by i.i.d. random points selected according to certain probability distributions, then, using polarity, we transfer this to the circumscribed model.