Symmetric Convex Domains Research Articles

Remarks on Soft Ball Packings in Dimensions 2 and 3

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

Remark on coisotropic Ekeland–Hofer–Zehnder capacity

In this paper, we give a counter-example to show that coisotropic Ekeland–Hofer–Zehnder capacities do not satisfy superadditivity for hyperplane cuts in higher dimension. Next, we show that the coisotropic Ekeland–Hofer–Zehnder capacity relative to Rn,k is not asymptotic equivalent to any normalized symplectic capacity in general case. But we show on the class of centrally symmetric convex domain in R2n, the coisotropic Ekeland–Hofer–Zehnder capacity is asymptotic equivalent to the Ekeland–Hofer–Zehnder capacity. Finally, we give some other results about the coisotropic Ekeland–Hofer–Zehnder capacity.

Regularity of Stationary Boltzmann Equation in Convex Domains

The higher regularity estimate has been a challenging question regarding the Boltzmann equation in bounded domains. Indeed, it is well-known to have “the non-existence of a second order derivative at the boundary” in Guo et al. (Invent Math 207:115–290, 2017) even for symmetric convex domains such as a disk or sphere. In this paper, we answer this question in the affirmative by constructing the \(C^{1,\beta }\) solutions away from the grazing boundary, for any \(\beta <1\), to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely.

On the moving plane method for singular solutions to semilinear elliptic equations

We consider positive weak solutions to −Δu=f(x,u) in Ω∖Γ with u=0 on ∂Ω. We prove symmetry and monotonicity properties of the solutions in symmetric convex domains via the moving plane method, under suitable assumptions on f and on the singular set Γ. With similar arguments we also consider the case when the domain is the whole space and the nonlinearity has at most critical growth.

The Set of Zeros of the Fourier Transform of the Characteristic Function of a Symmetric Convex Domain

We study the set N Ω of zeros of the Fourier transform of the characteristic function of a centrally symmetric convex domain Ω in the plane and the functional κ(Ω) = dist(0,N Ω). We prove that, if Ω is not a polygon, then N Ω is the countable union of disjoint analytic curves. We also show that, among centrally symmetric convex domains of a fixed area, there exists a domain where the functional κ attains the maximum. Bibliography: 4 titles.

Parallel X-ray tomography of convex domains as a search problem in two dimensions

In 1961, at A.M.S. Symposium on Convexity, P.C. Hammer proposed the following problem: how many X-ray pictures of a convex planar domain D must be taken to permit its exact reconstruction? Richard Gardner writes in his fundamental 2006 book [4] that X-rays in four different directions would do the job. The present paper points at the possibility that in certain asymptotical sense X-rays in only three different directions can be enough for approximate reconstruction of centrally symmetric convex domains. The accuracy of reconstruction would tend to become perfect in the limit, as the directions of the three X-rays change, all three converging to some given direction. The analysis leading to that conclusion is based on two lemmas of Section 1 and Pleijel type identity for parallel X-rays derived in Sections 2 and 3. These tools together supply a systemof two differential equations with respect to two unknown functions that describe the two branches of the domain boundary D. The system is easily resolved. The solution intended to provide a complete tomography reconstruction of D, happens however to depend on a two dimensional parameter, whose “real value” remains unknown. So tomography reconstruction of D becomes possible if a satisfactory approximation to that unknown “real value” can be found. In the last section a test procedure for the individual candidates for “approximate real value” of the parameter is described. A uniqueness theorem concerning tomography of circular discs is proved.

Transversals in Superdisjoint T(3)-families of Translates

Let K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. Property T means that a transversal exists for all members of the family. Two translates, K i and K j are said to be φ-disjoint, φ>0, if the concentric φ-enlarged copies of K i and K j are disjoint. It is well known that in a φ-disjoint family of congruent discs T(3)⇒T if \(\varphi>\sqrt{2}\), and \(T(3)\not\Rightarrow T\) if \(\varphi<\sqrt{2}\). In this note finite φ-disjoint T(3)-families of translates of an oval will be investigated.

Transversals with Residue in Moderately Overlapping T(k)-Families of Translates

AbstractLet K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. The integer k is the stabbing level of the family. Two translates Ki = K + ci and Kj = K + cj are said to be σ-disjoint if σK + ci and σK + cj are disjoint. A recent Helly-type result claims that for every σ &gt; 0 there exists an integer k(σ) such that if a family of σ-disjoint unit diameter discs has property T(k)|k ≥ k(σ), then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval K. The asymptotic behavior of k(σ) for σ → 0 is considered as well.Katchalski and Lewis proved the existence of a constant r such that for every pairwise disjoint family of translates of an oval K with property T(3) a straight line can be found meeting all but at most r members of the family. In the second part of the paper σ-disjoint families of translates of K are considered and the relation of σ and the residue r is investigated. The asymptotic behavior of r(σ) for σ → 0 is also discussed.

Weighted Poincare inequalities on symmetric convex domains

Let α ≥ 0, β ∈ R, 1 ≤ p ≤ q < ∞ with 1 - n / p + n / q , 1 - n + β / p + n + α / q ≥ 0. Let Ω be a bounded convex domain in ℝ n that is symmetric with respect to its center. Define p(x) = dist(x, Ω c ) = inf{|x - γ|: γ ∈ Ω c } and ρ α (E) = ∫ E ρ(x) α dx. Let f be a Lipschitz continuous function on Ω and f Ω,ρ α = ∫ Ω f(x)ρ(x) α dx/ρ α (Ω). We obtain the following weighted Poincare inequality: ∥f - f Ω,ρ α ∥L q ρα (Ω) ≤ Cη β/p-α/q |Ω| 1/q-1/p diam(Ω) 1-β/p+α/q ∥∇f∥ L p ρ β (Ω) where η is the eccentricity of Ω and C is a constant depending only on p, q, α, β, and the dimension n. Moreover, the exponent of η is sharp.

Asymptotic Uniqueness for a Biharmonic Equation with Nearly Critical Growth on Symmetric Convex Domains

We consider a biharmonic equation with the nearly critical Sobolev exponent under the Navier boundary condition on a smooth bounded, strictly convex domain of dimension N ≥ 5, which is symmetric with respect to the coordinate hyperplanes. We prove that the number of positive solutions of the above problem is exactly one when the nonlinear exponent is subcritical and

The simultaneous packing and covering constants in the plane

In 1950, C.A. Rogers introduced and studied two simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, these constants have attracted the interests of many authors because, besides their own geometric significance, they are closely related to the packing densities and the covering densities of the convex body, especially to the Minkowski–Hlawka theorem. However, so far our knowledge about them is still very limited. In this paper we will determine the optimal upper bound of the simultaneous packing and covering constants for two-dimensional centrally symmetric convex domains, and characterize the domains attaining this upper bound.

An inequality for potentials and the 'hot-spots' conjecture

Consider a nonnegative continuous potential V on the half disk D + = {z = x + iy: y > 0, |z| < 1} for which r 2 V(re iθ ) is nondecreasing as a function of r for every fixed 0 < θ < π. We prove an inequality for the distribution of the random variable ∫ τD+ 0 V(B s ) ds where B s is the Brownian motion reflected on the top portion of the boundary and killed on the lower portion and T D+ is its lifetime. This inequality, by the conformal invariance of Brownian motion, implies a result of Pascu [13] on hot-spots for certain symmetric convex domains.

Simultaneous Packing and Covering in the Euclidean Plane

In this article we study the simultaneous packing and covering constants of two-dimensional centrally symmetric convex domains. Besides an identity result between translative case and lattice case and a general upper bound, exact values for some special domains are determined. Similar to Mahler and Reinhardt’s result about packing densities, we show that the simultaneous packing and covering constant of an octagon is larger than that of a circle.

An inequality between Dirichlet and Neumann eigenvalues in a centrally symmetric domain

We prove an inequality between Dirichlet and Neumann eigenvalues of the Laplacian in a centrally symmetric Euclidean domain. David Jerison and Nikolai Nadirashvili conjectured in [1] (Conjecture 8.6) that, for a centrally symmetric convex domain, A2, the lowest nonzero Neumann eigenvalue of the Laplacian for an eigenfunction that is even with respect to the central symmetry x H-* -x is smaller than AL2, the lowest Dirichlet eigenvalue for an eigenfunction that is odd with respect this symmetry. Notice that this inequality does not follow from the general inequalities between the Dirichlet and the Neumann eigenvalues: it may happen that ,u2 is the second eigenvalue of the Dirichlet Laplacian, and A2 is, say, 100th eigenvalue of the Neumann Laplacian. It turned out that this conjecture, even in somewhat stronger form, has a rather simple proof. In this short paper, I present the proof. The technique is very similar to the one used in [2]. Theorem. Let Q be a centrally symmetric domain in TRn with C2-boundary F Q Secondary 58G25. 0)2000 American Mathematical Society 2057 This content downloaded from 207.46.13.118 on Sun, 11 Sep 2016 06:14:55 UTC All use subject to http://about.jstor.org/terms 2058 LEONID FRIEDLANDER The function v, (x) satisfies the equation AV, (x) + pu2vw (x) = O. so (2) ] Vv,12 dx = A2 ]v 12 dx?+ ]vW)(x) dA where 0/0v is differentiation in the outward normal direction, and dA is the surface element on F. Our goal is to show that (3) j dw jvw(x) v (x)dA < 0. Here dw is the standard measure on S`1 normalized in such a way that the total area of the sphere equals 1. Clearly, (3) implies that iVw(x)v< (x)dA <0 for some value of w, and then (1) would follow from (2). To prove (3), we will explicitly compute the integral on the left. First, on F, Vu = (0u/v so (4) VW (X)= 09U(X)(V a ), X Gr. a!, Here v(x) is the outward normal unit vector to F at x. In a neighborhood of F, we introduce local coordinates x = (Y', Yn), Y' (Y1i... , Yn-i), in such a way that Yn is the distance from a point x to F taken with the positive sign if the point lies outside Q, and taken with the negative sign otherwise; y' are the local coordinates on F of the point that is the closest to x. In this coordinate system, the Euclidean metric tensor equals g 9 /(Y' vYn) ?) where g' is a metric tensor on F that depends on Yn. In the y-coordinates, the Laplacian has the form 02 1&09log g' & ayn 2 9yn 0Yn where A' is the tangent Laplacian, and g' is the determinant of the covariant tensor g'(y', Yn). The function u(x) is an eigenfunction of the Dirichlet Laplacian, so one has Q2U 1 l V'u is a vector field in R' perpendicular to v. Then vw(y) = V'u(y' vy) W + a v W,

Separatrices splitting for Birkhoff's billiard in symmetric convex domain, closed to an ellipse.

An example of a convex domain on the plane with the phenomenon of the transversal intersection of separatrices of the corresponding billiard mapping is presented. This example is constructed as an analytic global symmetric perturbation of an ellipse and we investigate the global symmetric analytic perturbation of the integrable billiard mapping in the ellipse. We establish a theorem on the separatrices splitting of the perturbed billiard mapping and derive the asymptotic formulas for a homoclinic invariant as well as for a "principal" splitting angle of separatrices, arising from the hyperbolic fixed point of the mapping. (c) 1994 American Institute of Physics.

On maximum likelihood estimation of the center of a centrally symmetric convex domain

Given a sample from a uniform distribution on a centrally symmetric convex domain, a maximum likelihood estimator of its center is constructed, and its efficiency is discussed.

Finite coverings by translates of centrally symmetric convex domains

Bambah and Rogers proved that the area of a convex domain in the plane which can be covered byn translates of a given centrally symmetric convex domainC is at most (n−1)h(C)+a(C), whereh(C) denotes the area of the largest hexagon contained inC anda(C) stands for the area ofC. An improvement with a term of magnitude √n is given here. Our estimate implies that ifC is not a parallelogram, then any covering of any convex domain by at least 26 translates ofC is less economic than the thinnest covering of the whole plane by translates ofC.

Multiple Lattice Packings of Symmetric Convex Domains in the Plane

Si S est un domaine convexe centralement symetrique alors on a pour k=2, 3 et 4 S k (S)=kS 1 (S), ou S k (S) est un pavage

Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

A matrix representation of the primitive residue classes (mod 2𝑛)

Received by the editors March 27, 1956. 1 It has been conjectured that the symmetric convex domain in the plane of given lattice determinant and smallest area is bounded by line segments and osculating hyperbolae arcs. The discussion of such domains leads to systems of linear equations which have matrices just as considered in this note, and I found their group property when I tried to solve the equations.