Convex Bodies Research Articles

Sections of convex bodies and Hanner polytopes

Sections of convex bodies and Hanner polytopes

On surface of Apollonius of two ellipsoids

Apollonius defined the circle as the set of points that have a given ratio μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K'$$\\end{document}, which define two norms. Their unit balls are K and K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K'$$\\end{document}. The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K'$$\\end{document} with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere.

Positive bases, cones, Helly-type theorems

Abstract Assume that k ≤ d is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ d . We prove a Helly-type theorem: If for every subfamily 𝓒* ⊂ 𝓒 of size at most max{d + 1, 2(d – k + 1)} the set ⋂ 𝓒* contains a k-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.

The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group [formula omitted]

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K0⊆R2n, for t-graphs on a bounded domain Ω in the Heisenberg group Hn. When the prescribed datum H is constant and strictly smaller than the Finsler mean curvature of ∂Ω, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.

On convergence properties of the brain-state-in-a-convex-domain

Convergence in the presence of multiple equilibrium points is one of the most fundamental dynamical properties of a neural network (NN). Goal of the paper is to investigate convergence for the classic Brain-State-in-a-Box (BSB) NN model and some of its relevant generalizations named Brain-State-in-a-Convex-Body (BSCB). In particular, BSCB is a class of discrete-time NNs obtained by projecting a linear system onto a convex body of Rn. The main result in the paper is that the BSCB is convergent when the matrix of the linear system is symmetric and positive semidefinite or, otherwise, it is symmetric and the step size does not exceed a given bound depending only on the minimum eigenvalue of the matrix. This result generalizes previous results in the literature for BSB and BSCB and it gives a solid foundation for the use of BSCB as a content addressable memory (CAM). The result is proved via Lyapunov method and LaSalle’s Invariance Principle for discrete-time systems and by using some fundamental inequalities enjoyed by the projection operator onto convex sets as Bourbaki–Cheney–Goldstein inequality.

Sumset Estimates in Convex Geometry

Abstract Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

Further regularity results in P-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P-$$\\end{document}extremal setting

In our paper, Pluripotential theory and convex bodies: a Siciak-Zaharjuta theorem (Computational Methods and Function Theory (2020), 20(3-4), 571-590) by Bayraktar et al., we proved a regularity result of P-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P-$$\\end{document}extremal function. In this paper, we prove some other stronger regularity results analogue to those in the standard setting. Specifically we focus on proving the following proposition: Proposition: Let K=D¯⊂Cd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K={\\bar{D}}\\subset {\\mathbb {C}}^d$$\\end{document} be the closure of a bounded domain D with ∂D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial D$$\\end{document} of class C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document}. Then, for any Q∈Cα(K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q\\in {\\mathcal {C}}_\\alpha (K)$$\\end{document}, where Cα(K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_\\alpha (K)$$\\end{document} is the Hölder class α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} on K, we have VP,K,Q∈Cα(Cd)for a convex bodyP⊂(R+)d.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{P,K,Q}\\in {\\mathcal {C}}_\\alpha ({\\mathbb {C}}^d) \ ext { for a convex body } P\\subset (\\mathbb {R^+})^d.$$\\end{document}

Morse–Smale complexes on convex polyhedra

Motivated by applications in geomorphology, the aim of this paper is to extend Morse–Smale theory from smooth functions to the radial distance function (measured from an internal point), defining a convex polyhedron in 3-dimensional Euclidean space. The resulting polyhedral Morse–Smale complex may be regarded, on one hand, as a generalization of the Morse–Smale complex of the smooth radial distance function defining a smooth, convex body, on the other hand, it could be also regarded as a generalization of the Morse–Smale complex of the piecewise linear parallel distance function (measured from a plane), defining a polyhedral surface. Beyond similarities, our paper also highlights the marked differences between these three problems and it also relates our theory to other methods. Our work includes the design, implementation and testing of an explicit algorithm computing the Morse–Smale complex on a convex polyhedron.

Estimation of the centroid Banach–Mazur distance between planar convex bodies

UDC 514.18 We consider a version of the Banach–Mazur distance between two convex bodies C and D from E d with an additional requirement that their centroids coincide. We prove that for any two convex bodies C and D in E 2 .

From valuations on convex bodies to convex functions

From valuations on convex bodies to convex functions

Variational Worn Stones

We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.

Series expansions for random disc-polygons in smooth plane convex bodies

Abstract We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with $C^{k+1}_+$ -smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry.

Thermal accumulation effects modelling of investigate radial shaft seal wear and lubrication transition mechanism

A radial shaft seal (RSS) model is established to reveal mixed thermal lubrication (MTL) and wear performance at speeds ranging from 3000 r/min to 7000 r/min. The model takes into account the initial temperatures of the oil medium and the running time of the RSS to investigate the transition limit of lubrication. Then a thermal fluid-solid method was developed to simulate the lubrication performance of RSS, which shows lubrication transition boundaries through rough micro convex bodies for wear deformation. The RSS deforms wedge-shaped during the wear process and a localized fluid pressure is created on the convex bodies. Considering the wear deformation of the convex bodies due to the wedge effect, the micro performance of the RSS can fully describe the mechanism of the leakage rate. As the dynamic pressure decreases and the leakage rate increases with the temperature of the medium, the RSS lubrication gradually transitions from MTL to EHL. Thus, the initial medium temperature should be considered to maintain the RSS under the mixed lubrication regime with low friction.

On the Maximal Distance Between the Centers of Mass of a Planar Convex Body and Its Boundary

On the Maximal Distance Between the Centers of Mass of a Planar Convex Body and Its Boundary

Convex Bodies of Constant Width with Exponential Illumination Number

Convex Bodies of Constant Width with Exponential Illumination Number

Dynamics of bouncing convex body

We consider a dynamical system that describes a convex planer body with a smooth boundary, which falls under the influence of gravity and is elastically reflected upon hitting a wall. Under suitable restriction on the system’s parameters, the system is equivalent to a billiard problem within a cylinder. We prove that the dynamics of the system is intimately linked to a variational problem, corresponding to a twist map. Moreover, we utilise the established variational method to investigate the stability of trivial periodic orbits and near-integrable dynamics when this planar body closely approximates a disc with its centre of mass at the geometric centre. Numerical results illustrates our mathematical findings and highlight the system’s intricate dynamics.

Convex bodies associated to linear series of adelic divisors on quasi-projective varieties

AbstractIn this article we define and study convex bodies associated to linear series of adelic divisors over quasi-projective varieties that have been introduced recently by Yuan and Zhang. Restricting our attention to big adelic divisors, we deduce properties of volumes obtained by Yuan and Zhang using different convex geometric arguments. We go on to define augmented base loci and restricted volumes of adelic divisors following the works of Michael Nakamaye and develop a similar study using convex bodies to obtain analogous properties for restricted volumes. We closely follow methods developed originally by Lazarsfeld and Mustaţă

Position of the centroid of a planar convex body

It is well known that any planar convex body A permits to inscribe an affine-regular hexagon HA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_A$$\\end{document}. We prove that the centroid of A belongs to the homothetic image of HA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_A$$\\end{document} with ratio 421\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{4}{21}$$\\end{document} and the center in the center of HA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_A$$\\end{document}. This ratio cannot be decreased.

Differential vertebral body growth is maintained after vertebral body tethering surgery for idiopathic scoliosis: 4-year follow-up on 888 peri-apical vertebrae and 592 intervertebral discs.

To radiographically evaluate if vertebral body tethering (VBT) can maintain differential peri-apical vertebral growth at medium-term follow-up of 4years. A prospective, international, multicenter database was queried to identify idiopathic scoliosis patients treated with thoracic VBT. Concave vs. convex vertebral body height, vertebral wedging, and disc wedging of the 3 peri-apical vertebrae were measured by two independent observers at 5 timepoints (pre-operative to 4-year follow-up). 65 skeletally immature patients (60 female, mean 12.8years old, 21 with open triradiate cartilages) met inclusion criteria. Mean pre-operative maximum scoliosis of 50 ± 8° decreased significantly post-operatively to 27 ± 9° (p < 0.001), which remained stable at 4-year follow-up 30 ± 17° (p = 0.38 vs. post-operative). Mean instrumented scoliosis was 21 ± 14° at 4-year follow-up, which was significantly different than 4-year maximum scoliosis (p < 0.001). Mean pre-operative kyphosis of 30 ± 12° did not significantly change post-operatively (p = 1.0) and remained stable at 4-year follow-up (35 ± 18°; p = 0.05). Mean individual convex vertebral height increased from 17.7 ± 1.9mm to 19.8 ± 1.5mm (p < 0.001), while mean individual concave height increased from 14.8 ± 1.9mm to 17.6 ± 1.6mm (p < 0.001). Summing the peri-apical heights, the difference in height from pre-operative to 4-year follow-up was greater on the concave (8.3 ± 4.7mm) than on the convex side (6.2 ± 4.7mm) (p < 0.001). Mean individual vertebral wedging decreased from 6 ± 2° at pre-operative to 4 ± 2° at 4-year follow-up (p < 0.001). Mean total vertebral and disc wedging started at 29 ± 7° pre-operatively, decreased to 16 ± 6° at post-operative (p < 0.001), then further decreased to 14 ± 8° at 4-year follow-up (p < 0.001). Patients with open triradiate cartilages at the time of surgery had a larger height change over the 4years compared to those with closed triradiate cartilages (p < 0.001). Patients with idiopathic scoliosis treated with VBT demonstrated differential vertebral growth which was maintained at minimum 4-year follow-up. This effect was more pronounced in patients whose triradiate cartilages were open at the time of surgery. III.