Convex Bodies Research Articles

A Gauss curvature flow approach to the torsional Minkowski problem

This paper is devoted to reconsidering the Minkowski problem for torsional rigidity introduced by Colesanti-Fimiani [9]. Employing a Gauss curvature flow, we establish the new existence of smooth solutions of the torsional Minkowski problem under the premise of giving an initial convex body.

Hybrid lubrication model study of slip ring combination seal under the influence of frictional heat

This paper presents an experimental and theoretical investigation into the factors influencing the sealing performance of combined slip rings. Factoring in the frictional heat from micro convex bodies and convective heat exchange between the slip ring, oil film, and air, we formulate a thermoelastic flow hybrid lubrication model for the combined slip ring seal. This model calculates the distribution of oil film thickness, pressure, temperature, velocity, and viscosity in the sealing zone, drawing on the generalized average Reynolds equation and the transmembrane average energy equation, in conjunction with the heat conduction equation of the slip ring. Utilizing the Archard wear model, we also examine the wear characteristics of the combined slip ring seal, providing insights into seal wear under these conditions. The model enables an analysis of the interplay between parameters and their impact on seal performance. The method proposed accurately predicts friction and leakage in line with experimental data, thereby providing a theoretical foundation for further numerical investigation of seal characteristics.

Bounds on Area Involving Lattice Size

The lattice size of a lattice polygon $P$ is a combinatorial invariant of $P$ that was recently introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper, we establish sharp lower bounds on the area of plane convex bodies $P\subset\mathbb{R}^2$ that involve the lattice size of $P$. In particular, we improve bounds given by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons $P\subset\mathbb{R}^2$ of fixed lattice size ${\operatorname{ls_\square}}(P)$.

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1].To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.

Thermal contact of roughness macro and micro model to reveal seal leakage mechanism under mixed lubrication

Research on numerical methods for reciprocating seals, including revealing leakage rate mechanism under the elastic-hydraulic lubrication (EHL) regime are mature. However, micro convex bodies contact behaviors and their effect on lubrication transition mechanism are still used under the mixed lubrication (ML) and wear model insufficiently. Seal model with micro convex bodies deformation should be updated to reflect the lubrication transition from EHL to ML and reveal leakage failure mechanism. Reciprocating speed ranging from 0.1 m/s to 0.2 m/s and working pressure of 100 MPa can characterize the seal performance and failure of leakage mechanism especially in low speed and harsh film formation condition. The updated leakage rate formula and the Reynolds equation are modified considering contact factor and reveal leakage mechanism through macro and micro numerical method.

Chord measures in integral geometry and their Minkowski problems

AbstractTo the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.

A new genus of the Araucarian-feeding tribe Oxycraspedini (Coleoptera: Belidae) from Eocene Baltic amber

ABSTRACTThe new genus, Bojebenzonia n. gen. and its species B. karinae n. sp. from the tribe Oxycraspedini (Belidae: Oxycoryninae) are described from late Eocene Baltic amber. Bojebenzonia differs from the genus Oxycraspedus by its tibiae lacking apical spurs, subparallel pronotal sides, weakly convex body, and wide tibiae. A list of fossil species of the subfamily Oxycoryninae and a key to tribes and subtribes of the subfamily Oxycoryninae are given. It is suggested that the new genus developed feeding on either the family Araucariaceae or Doliostrobaceae. Trophic associations of extant Oxycoryninae are briefly discussed.

Thermohydrodynamic Lubrication Characteristics of Piston Rings in Diesel Engine Considering Transient Heat Transfer under the Parameterized Surface Texture of Cylinder Liners

The cylinder liner and piston ring form the most crucial friction pair in the diesel engine, contributing 35–40% of its overall friction losses. Recent research indicates that transient heat transfer significantly affects piston ring lubrication. However, the impact of such a transfer on varying surface textures and lubrication traits remains unclear. This paper takes the piston ring–cylinder liner of a certain diesel engine as the research object, which is based on a two-dimensional averaged Reynolds function and Greenwood–Tripp micro convex body contact model; establishes a numerical calculation model of the transient heat fluid lubrication characteristics of a vertical piston ring–cylinder liner assembly by combining the oil film thickness equation, energy equation, lubricating oil viscosity–temperature, and viscosity pressure characteristics; avoids large errors associated with assuming different temperature values for lubricants; and also uses the cylinder liner surface texturing technique to examine the effects of surface texturing on lubrication properties in the presence of transient thermal fluids. The findings indicate that employing transient thermal fluid for determining the mean value of the oil film temperature in isothermal lubrication calculations yields comparable values for minimum oil film thickness and frictional power consumption, while the friction power consumption calculated by the transient thermal fluid is slightly lower. The depth of the recesses on the surface of the cylinder liner should be minimized, while the radius of the texture should be maximized, taking into consideration the current circumstances. Compared with a cylindrical texture, a spherical texture achieves lower friction with good lubrication indexes.

Orientation-Dependent Chord Length Distribution in a Convex Quadrilateral

This work contributes to the research devoted to the recognition of a convex body by probabilistic characteristics of its lower-dimensional sections. In this paper, for any convex quadrilateral, five orientation-dependent characteristics are introduced and explicitly evaluated per direction. In terms of these characteristics, simple explicit representations of the orientation-dependent chord length distribution function and the covariogram are obtained not only for an arbitrary convex quadrilateral but also for any right prism based on it.

A Generalized Source Integral Equation for Enhanced Compression in Three Dimensions

An integral equation formulation, using generalized directional sources, for three-dimensional scattering by impenetrable and essentially convex bodies, is presented. This formulation increases the rank-deficiency of moment matrix blocks representing interactions between non-overlapping source and observer regions. Thereby, it enables enhanced low-rank approximation based matrix compression and the development of corresponding fast direct solvers. The directional sources are constructed by augmenting the conventional basis functions with spherical absorbing three-dimensional shields, on which auxiliary source distributions are defined. The bottleneck arising from the need to integrate over these distributions upon computing the modified Green’s function is removed by using efficient non-uniform sampling and tabulation of the modified Green’s function or its components, in a region-dependent manner. The formulation is studied and its favorable compressibility is demonstrated, for two fundamental types of compression strategies. The non-uniform sampling approach is employed also to facilitate the rank-revealing analysis of large off-diagonal blocks.

Cyclic inequalities for generalized mixed width-integrals

In this paper, we research the cyclic inequalities of generalized width-integral in convex geometry analysis. Based on previous studies, we use Hölder’s inequality and inverse Hölder’s inequality to study further the generalized width-integral inequalities of the convex body in Euclidean space, establish a series of cyclic inequalities, and give the condition of equality sign.

The limit point in the Jante’s law process has an absolutely continuous distribution

We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante’s law process. We consider a version of the model where the space of possible opinions is a convex body B in Rd. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jerónimo-Castro, Magazinov and Soberón which generalized a question posed by Dol'nikov. Let F1,F2,…,Fn be families of translates of a convex compact set K in the plane so that each two sets from distinct families intersect. We show that, for some j, ⋃i≠jFi can be pierced by at most 4 points. To do so, we use previous ideas from Gomez-Navarro and Roldán-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.

On the volume ratio of projections of convex bodies

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body K we show that there is another convex body L such that the volume ratio between any two projections of fixed rank of the bodies K and L is large. Namely, we prove that for every 1≤k≤n and for each convex body K⊂Rn there is a centrally symmetric body L⊂Rn such that for any two projections P,Q:Rn→Rn of rank k one hasvr(PK,QL)≥cmin⁡{kn1log⁡log⁡log⁡(nlog⁡(n)k),klog⁡(nlog⁡(n)k)}, where c>0 is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime k≥n2/3.

A note on the convex body isoperimetric conjecture in the plane

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area π \pi . In this note we confirm two cases of the conjecture: perturbations of the unit disk and domains bounded by curves symmetric to both coordinate axes and having exactly four vertices.

On the monotonicity of weighted perimeters of convex bodies

AbstractWe prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially‐weighted anisotropic densities.

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

Abstract In this paper we consider metric fillings of boundaries of convex bodies. We show that convex bodies are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As further applications of this result, we prove a variant of Lipschitz-volume rigidity for round spheres and answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.

Generalization Bounds in the Predict-Then-Optimize Framework

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation. Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.

A geometric approach to second-order differentiability of convex functions

We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by C 1 , 1 C^{1,1} convex functions and convex bodies.

Existence and approximation of densities of chord length- and cross section area distributions

In various stereological problems an $n$-dimensional convex body is intersected with an $(n-1)$-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the $(n-1)$-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.