On Hamiltonian Projective Billiards on Boundaries of Products of Convex Bodies

Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume v ex(K) for a general convex body K that applies in any space dimension. While our main interests concern the rotationally-averaged exclusion volume of a convex body with respect to another convex body, we also describe some results for the exclusion volumes for convex bodies with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume v ex(K)/v(K) among all convex bodies, whether randomly oriented or uniformly oriented, for any d, where v(K) is the volume of K. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any d with a large-d asymptotic scaling behavior of 22d /d 3/2, which is to be contrasted with the corresponding scaling of 2 d for the sphere. We present explicit formulas for quermassintegrals W 0(K), …, W d (K) for many different nonspherical convex bodies, including cubes, parallelepipeds, regular simplices, cross-polytopes, cylinders, spherocylinders, ellipsoids as well as lower-dimensional bodies, such as hyperplates and line segments. These results are utilized to determine the rotationally-averaged exclusion volume v ex(K) for these convex-body shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, we show that, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal v ex(K)/v(K) with a scaling behavior of 21.6618…d . Subsequently, we apply these results to determine the corresponding second virial coefficient B 2(K) of the aforementioned hard hyperparticles. Our results are also applied to compute estimates of the continuum percolation threshold η c derived previously by the authors for systems of identical overlapping convex bodies. We conjecture that overlapping spheres possess the maximal value of η c among all identical nonzero-volume convex overlapping bodies for d ⩾ 2, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of η c for d ⩾ 2.

In recent years P. C. Hammer's problem [8] of determining a convex body from its 'X-ray pictures' was investigated by Gardner and McMullen [4], Gardner [3], Falconer [2] and Volcic [15]. An earlier result is due to Giering [5]. An X-ray picture of a convex body in a direction may be identified with its Steiner symmetral in that direction. Some of these papers consider X-ray pictures taken from points not on the line at infinity, but here we are not concerned with that situation. Gardner and McMullen proved that there exist four directions such that the corresponding X-ray pictures distinguish between all convex bodies, and that no three directions can do this. Giering proved that, given a plane convex body K, there exist three directions depending on K, such that the corresponding X-ray pictures distinguish K from any other convex body. He has also shown that two directions are in general not enough. Convex bodies with the same X-ray pictures as a given one were called 'ghosts' in [14], in analogy with the ghost densities from computerized tomography [12]. It should be remembered that in the fundamental case of parallel rays from two orthogonal directions, besides a few triangular or quadrangular examples by Giering [6] and a rather obvious construction which basically interchanges two diagonally opposite, symmetrical pieces with two other diagonally opposite congruent pieces (diagonals of a rectangle), no deeper insight into the soul of a ghost of a convex body has been won. We are—as a consequence—far away from being able to characterize convex are not ghosts! (Note the equivalence between having and being a ghost!) Thus we In this situation, the question about the generic behaviour of convex bodies with regard to their ghosts appears interesting, but looks at a first glance, in view of the lack of knowledge described above, rather hopeless. However, in this paper we establish the validity of the (more comfortable?) assertion that most convex bodies are not ghosts! (Note the equivalence between having and being a ghost!) Thus we confirm a conjecture of the first author, motivated by the symmetries described above and also present in his examples from [14]. It is clear that the orthogonality of the two considered directions is unessential, because of the afrlne character of our problem. When we state it we do so just to fix the ideas. As a main open problem there remains the characterization of those convex bodies which are uniquely determined by two X-ray pictures. The analogous problem for measurable sets has been solved by Lorentz [11]. The space # of all convex curves in 1R, like the space & of all convex bodies in U, equipped with the Hausdorff distance S is a Baire space. 'Most ' means 'all, except those in a set of first category'. For a survey on properties of most convex bodies, see [16].

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel [Mathematika 52 (2005), 47{52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce a relative of the covering parameter called covering index, which turns out to have a number of nice properties. Intuitively, the covering index measures how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies and provides a useful upper bound for well-studied quantities like the illumination number, the illumination parameter, the vertex index and the covering parameter of a convex body. We obtain upper bounds on the covering index and investigate its optimizers. We show that the ane d-cubes minimize covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, we show that the covering index satises a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in innitely many instances. This together with a monotonicity property can be used to determine the covering index of innitely many convex bodies.

We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities. Such interpretations were first given by Paouris and Werner for symmetric convex bodies in the context of the L p L_p -centroid bodies. There, the relative entropies appear after performing second order expansions of certain expressions. Now, no symmetry assumptions are needed. Moreover, using the new bodies, already first order expansions make the relative entropies appear. Thus, these bodies detect “faster” details of the boundary of a convex body than the L p L_p -centroid bodies.

In this paper we study geometry of compact, not necessarily centrally symmetric, convex bodies in R. Over the years, local theory of Banach spaces developed many sophisticated methods to study centrally symmetric convex bodies; and already some time ago it became clear that many results, if valid for arbitrary convex bodies, may be of interest in other areas of mathematics. In recent years many results on non-centrally symmetric convex bodies were proved and a number of papers have been written (see e.g., [1], [8], [12], [18], [27], [28] among others). The present paper concentrates on random aspects of compact convex bodies and investigates some invariants fundamental in the local theory of Banach spaces, restricted to random sections and projections of such bodies. It turns out that, loosely speaking, such random operations kill the effect of non-symmetry in the sense that resulting estimates are very close to their centrally symmetric counterparts (this is despite the fact that random sections might be still far from being symmetric (see Section 5 below)). At the same time these estimates might be in a very essential way better than for general bodies. We are mostly interested in two directions. One is connected with so-called MM∗estimate, and related inequalities. For a centrally symmetric convex body K ⊂ R, an estimate M(K)M(K) ≤ c log n (see the definitions in Section 2 below) is an important technical tool intimately related to the Kconvexity constant. It follows by combining works by Lewis and by Figiel and Tomczak-Jaegermann, with deep results of Pisier on Rademacher projections (see e.g., [26]). Although the symmetry can be easily removed from the first two parts, Pisier’s argument use it in a very essential way. In Section 4 we show, in particular, that every convex body K has a position K1 (i.e., K1 = uK − a for some operator u and a ∈ R) such that a random projection, PK1, of dimension [n/2] satisfies M(PK1)M(K 1 ) ≤ C log n, where C is an absolute constant. Moreover, there exists a unitary operator u such that M(K1 +uK1)M(K 1 ) ≤ C log n. Our proof is based essentially on symmetric considerations, a non-symmetric part is reduced to classical facts and simple

According to a classical result of Grunbaum, the transversal number τ(ℱ) of any family ℱ of pairwise-intersecting translates or homothets of a convex body C in ℝd is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ(ℱ) to the packing number ν(ℱ) over all finite families ℱ of translates (resp. homothets) of a convex body C in ℝd . Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝd , and gave the first bounds on α(C) for convex bodies C in ℝd and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in ℝd , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝd for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.

Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume for a general convex body that applies in any space dimension, including both the rotationally-averaged exclusion volume and with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume $v_{ex}(K)/v(K)$ among all convex bodies, whether randomly oriented or uniformly oriented, for any $d$, where $v(K)$ is the volume of $K$. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any $d$ with a large-$d$ asymptotic scaling behavior of $2^{2d}/d^{3/2}$, which is to be contrasted with the scaling of $2^d$ for the sphere. We present explicit formulas for quermassintegrals for many nonspherical convex bodies as well as as well as lower-dimensional bodies. These results are utilized to determine the rotationally-averaged exclusion volume for these shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal dimensionless exclusion volume with a scaling behavior of $2^{1.6618\ldots d}$. We also determine the corresponding second virial coefficient $B_2(K)$ of the aforementioned hard hyperparticles and compute estimates of the continuum percolation threshold $\eta_c$ derived previously by the authors. We conjecture that overlapping spheres possess the maximal value of $\eta_c$ among all identical nonzero-volume convex overlapping bodies for $d \ge 2$, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of $\eta_c$ for $d\ge 2$.

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function [Formula: see text] one has [Formula: see text] where [Formula: see text]. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.

We consider the problem of the approximation of regular convex bodies in ℝ d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is \(\frac{1}{n}\). Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then \(\frac{1}{n}\) is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C 2-smoothness the rate of approximation is \(O(\frac{1}{n^{2}})\).

ABSTRACTA correction is made to a previous paper(2) on clumps formed by random placing of laminae, a useful property of such placings is deduced and formulae are found for clumps formed when convex bodies are randomly placed in space and for clumps formed by the orthogonal projections of such placings.

Smoothness of Minkowski sum and generic rotations

Let n ≥ 3 and let K ⊂ ℝ n be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p, the section Π ∩ K has an (n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K, p is in the corresponding (n – 2)-plane of symmetry, then we call p a revolution point of K. We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝ n is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in [7]. We also show that if p is a Larman point of K ⊂ ℝ3 and there exists a line L such that p ∉ L and, for every plane Π passing through p, the line of symmetry of the section Π ∩ K intersects L, then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝ n with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D, then K is a body of revolution with axis D.

It is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the Legendre ellipsoid of classical mechanics. Sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid. These inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid. Among other things, this leads to a new approach to establishing Ball’s maximal shadows conjecture (for symmetric convex bodies). Corresponding to each origin–symmetric convex (or more general) subset of Euclidean n-space, R, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1-dimensional subspace of R. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See Milman and Pajor [MPa1, MPa2], Lindenstrauss and Milman [LiM] and Leichtweis [Le] for some historical references. It has slowly come to be recognized that along side the Brunn-Minkowski theory there is a dual theory. The nature of the duality between this dual Brunn-Minkowski theory and the Brunn-Minkowski theory is subtle and not yet understood. It is easily seen that the Legendre (and Binet) ellipsoid is an object of this dual BrunnMinkowski theory. This observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical Legendre ellipsoid in the Brunn-Minkowski theory. It is the aim of this paper to demonstrate the existence of precisely this dual object. In retrospect, one may well wonder why Research supported, in part, by NSF Grant DMS–9803261 Typeset by AMS-TEX 1 2 A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES the new ellipsoid presented in this note was not discovered long ago. The simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of L2-curvature in hand. However, the Brunn-Minkowski theory was only recently extended to incorporate the new notion of Lp-curvature (see [L2], [L3]). A positive definite n × n real symmetric matrix A generates an ellipsoid, (A), in R, defined by (A) = {x ∈ R : x·Ax ≤ 1}, where x·Ax denotes the standard inner product of x and Ax in R. Associated with a star-shaped (about the origin) set K ⊂ R is its Legedre ellipsoid, Γ2K, which is generated by the matrix [mij(K)] where mij(K) = n+ 2 V (K) ∫ K (ei ·x)(ej ·x) dx, with e1, . . . , en denoting the standard basis for R and V (K) denoting the ndimensional volume of K. We will associate a new ellipsoid Γ−2K with each convex body K ⊂ R. One approach to defining Γ−2K without introducing new notation is to first define it for polytopes and then use approximation (with respect to the Hausdorff metric) to extend the definition to all convex bodies. Suppose P ⊂ R is a polytope that contains the origin in its interior. Let u1, . . . , uN denote the outer unit normals to the faces of P , let a1, . . . , aN denote the areas (i.e., (n − 1)-dimensional volumes) of the corresponding faces and let h1, . . . , hN denote the distances from the origin to the corresponding faces. The ellipsoid Γ−2P is generated by the matrix [mij(P )] where mij(P ) = 1 V (P ) N ∑ l=1 al hl (ei ·ul)(ej ·ul). An alternate definition of the operator Γ−2 will be given after additional notation is introduced. The easily established affine nature of the operator Γ2 is formally stated in: Lemma 1. If K ⊂ R is star shaped about the origin, then for each φ ∈ GL(n),

According to a classical result of Grünbaum, the transversal number \(\tau({\mathcal F})\) of any family \({\mathcal F}\) of pairwise-intersecting translates or homothets of a convex body C in ℝd is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number \(\tau({\mathcal F})\) to the packing number \(\nu({\mathcal F})\) over all families \({\mathcal F}\) of translates (resp. homothets) of a convex body C in ℝd. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝd, and gave the first bounds on α(C) for convex bodies C in ℝd and on β(C) for convex bodies C in the plane. In this paper, we show that β(C) is also bounded by a function of d for any convex body C in ℝd, and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝd for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.KeywordsLattice PointConvex BodyLattice PackingSymmetric ConvexSymmetric Convex BodyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In order to study the airflow distribution in irregular cross-section roadway, the numerical simulation method were carried out to simulate the influence of hemispheres with radius of 100mm, 200mm, 300mm, and 400mm placed in different positions in the roadway on the distribution of “characteristic rings”. The results showed that concave body is not the key factor influencing the distribution of “characteristic ring” in ventilation roadway. Convex body is the key factor influencing the distribution of “characteristic ring”, the greater the ventilation velocity, the smaller the velocity gradient in the stable zone and the velocities in the stable area are more closely to the average velocity. As the increase of ventilation velocity, the velocity gradient is larger in transition zone. The size and position of the convex and concave bodies are not the key factors influencing the velocity distribution on the central axis of cross-section.