Symmetric Convex Sets Research Articles

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces

Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces

Improved log-concavity for rotationally invariant measures of symmetric convex sets

Improved log-concavity for rotationally invariant measures of symmetric convex sets

On the Gardner-Zvavitch conjecture: Symmetry in inequalities of Brunn-Minkowski type

In this paper, we study the conjecture of Gardner and Zvavitch from [22], which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of symmetric convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L, and λ∈[0,1],γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n. More generally, this inequality holds for convex sets containing the origin. Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p>0, with respect to the addition of symmetric convex sets.

Symmetric convex sets with minimal Gaussian surface area

Let $\Omega\subset\Bbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_x$ be the second fundamental form of $\partial\Omega$ at $x$, i.e., $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in\partial\Omega$. For any $x=(x_1,\ldots,x_{n+1})\in\Bbb{R}^{n+1}$, let $\gamma_n(x)=(2\pi)^{-n/2}e^{-(x_1^2+\cdots+x_{n+1}^2)/2}$. Let $\|A\|^{2}$ be the sum of the squares of the entries of $A$, and let $\|A\|_{2\to 2}$ denote the $\ell_{2}$ operator norm of $A$.

The dimensional Brunn–Minkowski inequality in Gauss space

Let γn be the standard Gaussian measure on Rn. We prove that for every symmetric convex sets K,L in Rn and every λ∈(0,1),γn(λK+(1−λ)L)1n⩾λγn(K)1n+(1−λ)γn(L)1n, thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn–Minkowski inequality for the Lebesgue measure. We also show that, for a fixed λ∈(0,1), equality is attained if and only if K=L.

Faster integer multiplication using short lattice vectors

We prove that $n$-bit integers may be multiplied in $O(n \log n \, 4^{\log^* n})$ bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is unconditional, and depends in an essential way on Minkowski's theorem concerning lattice vectors in symmetric convex sets.

Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constantC(n) =Cn7depending on the space dimensionnin both inequalities is due to Segal [A. Segal,Lect. Notes Math., Springer, Heidelberg2050(2012) 381–391]. We improve that constant toCn6for convex sets and toCn5for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the formCn2,i.e., quadratic inn. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets

This survey is based on a series of lectures given by the authors at the working seminar Convexite et Probabilites at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal inequalities associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances. The series focused on proving for these maximal functions inequalities in $L^p(\mathbb{R}^n)$ with bounds independent of the dimension $n$, for all $p \in (1, +\infty]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and Muller during the period 1986--1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with negative results for the weak type $(1, 1)$ constant, due to Aldaz, Aubrun and Iakovlev--Stromberg between 2009 and 2013.

Entanglement and nonlocality in diagonal symmetric states of N qubits

We analyze entanglement and nonlocal properties of the convex set of symmetric $N$-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

In this paper we present new versions of the classical Brunn-Minkowski inequality for dierent classes of measures and sets. We show that the inequality (A + (1 )B) 1=n (A) 1=n + (1 ) (B) 1=n holds true for an unconditional product measure with decreasing density and a pair of unconditional convex bodies A;B R n . We also show that the above inequality is true for any unconditional logconcave measure and unconditional convex bodiesA;B R n . Finally, we prove that the inequality is true for a symmetric log-concave measure and a pair of symmetric convex sets A;B R 2 , which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed in [13]. In addition, we deduce the 1=n-concavity of the parallel volume t7! (A +tB), Brunn’s type theorem and certain analogues of Minkowski first inequality.

Constructive Discrepancy Minimization for Convex Sets

A classical theorem of Spencer shows that any set system with $n$ sets and $n$ elements admits a coloring of discrepancy $O(\sqrt{n})$. Recent exciting work of Bansal, Lovett, and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett--Meka algorithm finds a half integral point in any “large enough” polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set $K$ with Gaussian measure at least $e^{-n/500}$, the following algorithm finds a point $y \in K \cap [-1,1]^n$ with $\Omega(n)$ coordinates in $\pm 1$: (1) take a random Gaussian vector $x$; (2) compute the point $y$ in $K \cap [-1,1]^n$ that is closest to $x$; (3) return $y$. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a theorem of Gluskin and Giannopoulos.

On minimal trajectories for mobile sampling of bandlimited fields

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focusing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.

Helly type theorems for the sum of vectors in a normed plane

The main results here are two Helly type theorems for the sum of (at most) unit vectors in a normed plane. Also, we give a new characterization of centrally symmetric convex sets in the plane.

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume $ {{4} \left/ {{\sqrt[3]{3} < 2.7734451}} \right.} $ . This fact implies that space contains a lattice packing of translates of K with density $ {{{\sqrt[3]{3}}} \left/ {{4 > 0.36056}} \right.} $ . Furthermore, K is contained in a parallelepiped of volume $ \frac{4}{3}{\left( {2 + \sqrt {3} } \right)^{{{2} \left/ {3} \right.}}} < 3.2080203 $ . Bibliography: 6 titles.

If you can hide behind it, can you hide inside it?

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. Does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? A compact convex set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of K, it must follow that L contains a translate of K. It is shown that, for 1 ≤ d ≤ n− 1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in Rn is d-reliable if and only if, for all m ≥ d+2, no m unit normals to regular boundary points of L form the outer unit normals of an (m− 1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in Rn is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in Rn is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable. As a result of (5) above, the only reliable centrally symmetric covers in R3 from the perspective of 2-dimensional shadows are the affine convex cylinders (prisms). However, in dimensions greater than 3, it is shown that 3decomposability is only sufficient, and not necessary, for L to cover reliably with respect to 3-shadows, even when L is assumed to be centrally symmetric. Consider two compact convex subsetsK and L of n-dimensional Euclidean space. Suppose that, for a given dimension 1 ≤ d 0 is sufficiently small), but no longer fits inside D. The same construction works for any set K inscribed in D and having strictly less than unit diameter. Another Received by the editors June 4, 2009. 2000 Mathematics Subject Classification. Primary 52A20. c ©2011 Daniel A. Klain

Covering shadows with a smaller volume

For n ⩾ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection L u onto the subspace u ⊥ contains a translate of K u for every direction u, while the volumes of K and L satisfy V n ( K ) > V n ( L ) . A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection L ξ onto a k-dimensional subspace ξ contains a translate of K ξ , while the mth intrinsic volumes of K and L satisfy V m ( K ) > V m ( L ) for all m > k . For each k = 1 , … , n , we then define the collection C n , k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms). It is subsequently shown that, if L ∈ C n , k , and if the orthogonal projection L ξ contains a translate of K ξ for every k-dimensional subspace ξ of R n , then V n ( K ) ⩽ V n ( L ) . The families C n , k , called k-cylinder bodies of R n , form a strictly increasing chain C n , 1 ⊂ C n , 2 ⊂ ⋯ ⊂ C n , n − 1 ⊂ C n , n , where C n , 1 is precisely the collection of centrally symmetric compact convex sets in R n , while C n , n is the collection of all compact convex sets in R n . Members of each family C n , k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C n , k are shown to satisfy certain geometric inequalities. Related open questions are also posed.

On Covering Problems of Rado

T. Rado conjectured in 1928 that if ${\mathcal S}$ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ${\mathcal I}\subseteq$ ${\mathcal S}$ of pairwise disjoint squares, such that ${\mathcal I}$ covers at least 1/4 of the area covered by ${\mathcal S}$. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ${\mathcal S}$. The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (1949, 1951 and 1968) on this subject. After 45 years (in 1973), Ajtai came up with a surprising example disproving T. Rado's conjecture. We revisit Rado's problem and present improved upper and lower bounds for squares, disks, convex sets, centrally symmetric convex sets, and others, as well as algorithmic solutions to these variants of the problem.

Distance sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ⊂ ℤ+ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials.Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [−q, q]2) ≦ C(K) q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L 2(K) does not possess an orthogonal basis of exponentials.

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S $${S \subset\mathbb{R}^n}$$ and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point $${x \in S}$$, let sym(x,S) denote the symmetry value of x in S: $${{\bf sym}(x,S):= max\{\alpha \ge 0 : x+\alpha(x-y) \in S {\rm for every} y \in S\}}$$, which essentially measures how symmetric S is about the point x, and define $${\bf sym}({\it S}):= \max_{x\in S} \, {\bf sym}({\it x,S})$$ x* is called a symmetry point of S if x* achieves the above maximum. The set S is a symmetric set if sym(S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym(S) and/or sym(x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym(x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the Lowner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory.