Symmetric Convex Sets Research Articles

On volume product inequalities for convex sets

The volume of the polar body of a symmetric convex set K of R d is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the L p -Blaschke-Santalo inequality for two-dimensional convex sets.

Inequalities for integral means over symmetric sets

We prove that the integral of n functions over a symmetric set L in R n , with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in R n . The average of such a function on E is maximized by the average over the symmetric set E * .

Mean value inequalities for convex and star-shaped sets

Let \(A \subset B \subset \mathbb{R}^{n} \) and let 0 be an interior point of A. It is shown that if A and B are compact star-shaped sets at 0 and if 0 is their common barycenter, then there is a positive number κ such that for every 0 < λ ≤ κ, the set λ A is convexly majorized by B. If, in addition, B is a symmetric (i.e. −B = B) convex set, it is shown that there is a universal positive constant κn, which depends only on the dimension n, such that for every symmetric convex set A, which satisfies the relation A ⊂ κ nB and if A and B have the same barycenter, then A is convexly majorized by B.

Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S ′ that contains C. More precisely, for any ɛ > 0 , we find an axially symmetric convex polygon Q ⊂ C with area | Q | > ( 1 − ɛ ) | S | and we find an axially symmetric convex polygon Q ′ containing C with area | Q ′ | < ( 1 + ɛ ) | S ′ | . We assume that C is given in a data structure that allows to answer the following two types of query in time T C : given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C ∩ ℓ . For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T C = O ( log n ) . Then we can find Q and Q ′ in time O ( ɛ − 1 / 2 T C + ɛ − 3 / 2 ) . Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O ( ɛ − 1 / 2 T C ) .

The shattering dimension of sets of linear functionals

We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization techniques and tail estimates of norms, viewed as random variables on Euclidean spheres. The estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions, a fact which was the original motivation of this study. Although this can probably be done in a more traditional manner, we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem.

The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems

We show that given a symmetric convex set K⊂ R d , the function t→γ(e tK) is log-concave on R , where γ denotes the standard d-dimensional Gaussian measure. We also comment on the extension of this property to unconditional log-concave measures and sets, and on the complex case.

On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

Pitt's conjecture (1977) that P( A ∩ B) ≥ P( A) P( B) under the Nn (0, In) distribution of X, where A, B are symmetric convex sets in IRn still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRn. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitt's conjecture.

Extension theorems for functional equations with bisymmetric operations

In this paper we deal with the extension of the following functional equation¶¶\( f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) \), (*)¶ where M is a k-variable operation on the image space Y, m1,..., mk are binary operations on X, \( K \subset X \) is closed under the operations m1,..., mk, and \( f : K \rightarrow Y \) is considered as an unknown function.¶ The main result of this paper states that if the operations m1,..., mk, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m1,..., mk)-affine hull K* of K, such that (*) holds over K*. (The set K* is defined as the smallest subset of X that contains K and is (m1,..., mk)-affine, i.e., if \( x \in X \), and there exists \( y \in K^* \) such that \( m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* \) then \( x \in K^* \)). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.

Optimization of Shape In Continuum Percolation

We consider a version of the Boolean (or Poisson blob) continuum percolation model where, at each point of a Poisson point process in the Euclidean plane with intensity $\lambda$, a copy of a given compact convex set $A$ with fixed rotation is placed. To each $A$ we associate a critical value $\lambda_c (A)$ which is the infimum of intensities $\lambda$ for which the occupied component contains an unbounded connected component. It is shown that $\min\{\lambda_c(A):A \text{ convex of area } a\}$ is attained if $A$ is any triangle of area $a$ and $\max\{\lambda_c(A):A \text{ convex of area } a\}$ is attained for some centrally symmetric convex set $A$ of area $a$. It turns out that the key result, which is also of independent interest, is a strong version of the difference­body inequality for convex sets in the plane. In the plane, the difference­body inequality states that for any compact convex set $A, 4\mu (A) \le \mu (A \oplus \check{A}) \le 6\mu (A)$ with equality to the left iff $A$ is centrally symmetric and with equality to the right iff $A$ is a triangle. Here $\mu$ denotes area and $A \oplus \check{A}$ is the difference­body of $A$. We strengthen this to the following result: For any compact convex set $A$ there exist a centrally symmetric convex set $C$ and a triangle $T$ such that $\mu(C) = \mu(T) = \mu(A)$ and $C \oplus \check{C} \subseteq A \oplus \check{A} \subseteq T \oplus \check{T}$ with equality to the left iff $A$ is centrally symmetric and to the right iff $A$ is a triangle.

A Particular Case of Correlation Inequality for the Gaussian Measure

Our purpose is to prove a particular case of a conjecture concerning the Gaussian measure of the intersection of two symmetric convex sets of $\mathbb{R}^n$. This conjecture states that the measure of the intersection is greater or equal to the product of the measures. In this paper, we prove the inequality when one of the two convex sets is a symmetric ellipsoid and the other one is simply symmetric. The general case is still open.

Even valuations on convex bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.

Une inégalité de décorrélation pour la mesure gaussienne

The purpose of this Note is to prove a particular case of an inequality concerning the Gaussian measure of the intersection of two symmetric convex sets of n . More precisely, we prove the inequality when one of the two convex is a symmetric ellipsoid and the other one is simply symmetric. The general case is still open.

Geometric orderings of intersecting translates and their applications

Let S be a family of n translates of a centrally symmetric convex set in the plane such that the intersection of any k of its members is empty. The number of geometric orderings induced by the line transversals of S is at most 8 k − 9. This bound can be used to prove that S has a line transversal if and only if every 64 k 2 − 122 k + 59 members of S have a line transversal. The bound can also be used to develop an efficient algorithm computing a line transversal of S in O( nk log k) time, if one exists.

Inequalities of correlation type for symmetric stable random vectors

We point out a certain class of functions f and g for which random variables f( X 1, …, X m ) and g( X m + 1 , …, X k ) are non-negatively correlated for any symmetric jointly stable random variables X i . We also show another result that is related to the correlation problem for Gaussian measures of symmetric convex sets.

Some inequalities for symmetric convex sets with applications

Under appropriate conditions the probability of a convex symmetric set decreases as the spread or scatter of the distribution increases. This paper studies the conditions when the random vector has a symmetric unimodal distribution.

A NEW APPROACH TO FINDING OPTIMAL LINEAR SCHEDULES FOR UNIFORM DEPENDENCE ALGORITHMS†

A uniform dependence algorithm can be represented by an index set of index points and a finite set of data dependence vectors. Usually, the convex hull R of the index set is a nondegenerated convex polytope in R n, In this paper, we show that finding an optimal linear schedule for a uniform dependence algorithm with an arbitrary bounded convex index set is equivalent to finding a vector of smallest norm, where the vector norm is defined on a symmetric convex set R∗ (which is the dual of difference body R − R). A linear programming problem is derived for finding the smallest vector. This problem can be solved in the empirical average time complexity O(36n 3 + 12an 2 + a), where a is the number of the linear inequalities defining the convex hull R and n is the dimension of the index set. This time complexity is better than those of the existing methods.

Monotonicity Properties of the Power Functions of Likelihood Ratio Tests for Normal Mean Hypotheses Constrained by a Linear Space and a Cone

Anderson studied the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric convex set. Restricting attention to elliptically contoured, unimodal densities, Mukerjee, Robertson and Wright weakened the assumption of symmetry on the set and obtained monotonicity properties of power functions, including unbiasedness, for some likelihood ratio tests in order restricted inference for the variance-known case. For elliptically contoured, unimodal densities, we weaken the assumption of convexity to obtain similar results in the case of unknown variances. The results apply to situations in which the null hypothesis is a linear space and the alternative is a closed, convex cone.

Majorants and Minorants for Elliptic Measures on [formula omitted

Let μ(· ; Σ, G1) and μ(· ; Ω, G2) be elliptically contoured measures on Rk centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G1, G2). Elliptical measures vm(·) and vM(·), depending on (Σ, Ω, G1, G2), are constructed such that Vm(C) ≤ {μ(C; Σ, G1), μ(C; Ω, G2)} for every symmetric convex set C ⊂ Rk with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.

Structure of homothetic linearly separable sets in n-dimensional euclidean space. I

We examine the structure of a finite collection of nonintersecting homothetic centrally symmetric convex sets in Rn that form a system of linearly separable sets.

Concentration inequalities for multivariate distributions: I. multivariate normal distributions

Let X ∼ N p (0, Σ), the p-variate normal distribution with mean 0 and positive definite covariance matrix Σ. Anderson (1955) showed that if Σ 2 − Σ 1 is positive semidefinite then P Σ 1 ( C) ⩾ P Σ 2 ( C) for every centrally symmetric (− C = C) convex set C⊆ R p . Fefferman, Jodeit and Perlman (1972) extended this result to elliptically contoured distributions. In the present study similar multivariate concentration inequalities are investigated for convex sets C that satisfy a more general symmetry condition, namely invariance under a group G of orthogonal transformations on R p , as well as for non-convex sets C that are monotonically decreasing with respect to a pre-ordering determined by G. Both new results and counterexamples are presented. Concentration inequalities may be used to convert classical efficiency comparisons, expressed in terms of covariance matrices, into comparisons of probabilities of multivariate regions.