Symmetric Convex Research Articles

Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures

We strengthen the volume inequalities for $L_p$ zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case $p=\infty$ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for $L_p$ zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe’s measure transportation proofs of the rank one case of the geometric Brascamp–Lieb and reverse Brascamp–Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp–Lieb and reverse Brascamp–Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky–Rogers lemma.

Subspace concentration of dual curvature measures of symmetric convex bodies

We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body. © 2018 International Press of Boston, Inc. All Rights Reserved.

A Subclass of $$\alpha $$ α -Convex Functions with Respect to (2j, k)-Symmetric Conjugate Points

The theory of (j, k)-symmetric functions has many applications in the investigation of fixed points, estimation of the absolute values of some integrals and in obtaining results of the type of Cartan’s uniqueness theorem. The concept of $$\left( 2j,k\right) $$ -symmetric functions extends the idea of even, odd, k, 2k, $$\left( j,k\right) $$ -symmetric and conjugate functions. In this paper, we introduce a new class $$\mathcal {M}_{\mathrm {SCP}} ^{j,k}\left( \alpha ,\eta ,\delta \right) $$ of analytic functions using the notion of $$\left( 2j,k\right) $$ -symmetric conjugate points. It unifies the classes $$\mathcal {S}_{\mathrm {SCP}}^{j,k}\left( \eta ,\delta \right) $$ and $$\mathcal {C}_{\mathrm {SCP}}^{j,k}\left( \eta ,\delta \right) $$ of starlike functions with respect to symmetric conjugate points and convex functions with respect to symmetric conjugate points, respectively. We also derive some inclusion results, integral representations and convolution conditions for functions belonging to the general function class $$\mathcal {M}_{\mathrm {SCP} }^{j,k}\left( \alpha ,\eta ,\delta \right) $$ . The various results presented in this paper may apply to yield the corresponding (new or known) results for a number of simpler known classes.

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

On the Approximation of Convex Bodies by Ellipses with Respect to the Symmetric Difference Metric

Given a centrally symmetric convex body \(K \subset \mathbb {R}^d\) and a positive number \(\lambda \), we consider, among all ellipsoids \(E \subset \mathbb {R}^d\) of volume \(\lambda \), those that best approximate K with respect to the symmetric difference metric, or equivalently that maximize the volume of \(E\cap K\): these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that \(\lambda \) is between the volumes of the John and the Loewner ellipsoids of K) is open in general. We provide a positive answer to this question in dimension \(d=2\). Therefore we obtain a continuous 1-parameter family of ellipses interpolating between the John and the Loewner ellipses of K. In order to prove uniqueness, we show that the area \(I_K(E)\) of the intersection \(K \cap E\) is a strictly quasiconcave function of the ellipse E, with respect to the natural affine structure on the set of ellipses of area \(\lambda \). The proof relies on smoothening K, putting it in general position, and obtaining uniform estimates for certain derivatives of the function \(I_K(\mathord {\cdot })\). Finally, we provide a characterization of maximal intersection positions, that is, the situation where the MI ellipse of K is the unit disk, under the assumption that the two boundaries are transverse.

Dynamic collective bargaining and labor adjustment costs

Collective bargaining between a trade union and a firm is analyzed within the framework of a monopoly union model as a dynamic Stackelberg game. Adjustment costs for the firm are comprised of the standard symmetric convex costs plus a wage-dependent element. Indeed, hiring costs can turn into benefits assuming wage discrimination against new entrants. The union also bears increasing marginal costs in the number of layoff workers and decreasing marginal benefits in the number of new entrants. Starting from a baseline scenario with instantaneous adjustment, we characterize the conditions under which the adjustment costs for the firm, or for the union, lead to higher employment and lower wages or vice versa. More generally, these adjustment costs, when they affect both the union and the firm, are generally detrimental to employment. However, the standard symmetric element of the adjustment costs for the firm positively affects employment, even with lower wages. Finally, if hiring and firing costs are defined separately, then hiring and firing could take place simultaneously if the wage discrimination towards new entrants is strong, because the firm would agree to pay the costs of firing incumbent employees, in order to enjoy wage savings from new entrants.

On Mergers in a Stackelberg Market with Asymmetric Convex Costs

This paper analyzes profitability, the incentives to free ride and the price effects of horizontal mergers in a Stackelberg market with an efficient leader and a group of inefficient followers when costs are convex. It is shown that a leader-follower merger is always profitable even though the free-riding incentives may reappear in situations where it would be absent with symmetric convex costs. On the contrary, a merger of two followers is only profitable if they are inefficient enough since in this case the free-riding behavior of the non-merging followers is more than compensated by the efficient reallocation of outputs derived from the merger. Finally, we also show that the mergers considered generally increase price except when the leader and a moderately inefficient follower merge.

Minimizing the mean projections of finite $$\rho $$ ρ -separable packings

A packing of translates of a convex body in the d-dimensional Euclidean space \({{\mathrm{\mathbb {E}}}}^d\) is said to be totally separable if any two packing elements can be separated by a hyperplane of \(\mathbb {E}^{d}\) disjoint from the interior of every packing element. We call the packing \(\mathcal P\) of translates of a centrally symmetric convex body \(\mathbf {C}\) in \({{\mathrm{\mathbb {E}}}}^d\) a \(\rho \)-separable packing for given \(\rho \ge 1\) if in every ball concentric to a packing element of \(\mathcal P\) having radius \(\rho \) (measured in the norm generated by \(\mathbf {C}\)) the corresponding sub-packing of \(\mathcal P\) is totally separable. The main result of this paper is the following theorem. Consider the convex hull \({{\mathrm{\mathbf {Q}}}}\) of n non-overlapping translates of an arbitrary centrally symmetric convex body \(\mathbf {C}\) forming a \(\rho \)-separable packing in \({{\mathrm{\mathbb {E}}}}^d\) with n being sufficiently large for given \(\rho \ge 1\). If \({{\mathrm{\mathbf {Q}}}}\) has minimal mean i-dimensional projection for given i with \(1\le i<d\), then \({{\mathrm{\mathbf {Q}}}}\) is approximately a d-dimensional ball. This extends a theorem of Boroczky (Monatsh Math 118:41–54, 1994) from translative packings to \(\rho \)-separable translative packings for \(\rho \ge 1\).

Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

Abstract Let n ≥ 2 {n\geq 2} be an integer, P = diag ⁢ ( - I n - κ , I κ , - I n - κ , I κ ) {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer κ ∈ [ 0 , n ] {\kappa\in[0,n]} , and let Σ ⊂ ℝ 2 ⁢ n {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., x ∈ Σ {x\in\Sigma} implies P ⁢ x ∈ Σ {Px\in\Sigma} , and ( r , R ) {(r,R)} -pinched. In this paper, we prove that when R / r &lt; 5 / 3 {{R/r}&lt;\sqrt{5/3}} and 0 ≤ κ ≤ [ n - 1 2 ] {0\leq\kappa\leq[\frac{n-1}{2}]} , there exist at least E ⁢ ( n - 2 ⁢ κ - 1 2 ) + E ⁢ ( n - 2 ⁢ κ - 1 3 ) {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when R / r &lt; 3 / 2 {{R/r}&lt;\sqrt{3/2}} , [ n + 1 2 ] ≤ κ ≤ n {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly n P-invariant closed characteristics, then there exist at least 2 ⁢ E ⁢ ( 2 ⁢ κ - n - 1 4 ) + E ⁢ ( n - κ - 1 3 ) {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function E ⁢ ( a ) {E(a)} is defined as E ⁢ ( a ) = min ⁡ { k ∈ ℤ ∣ k ≥ a } {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any a ∈ ℝ {a\in\mathbb{R}} .

Axes for symmetric convex curves

Curves are given in polar coordinates (r, θ)by equations of the form r = f (θ), where for f (θ) &gt; 0 all θ. Consider curves which are symmetric about the origin O, so that, f(θ + π) = f (θ) for all θ. For such a curve, its interior is the set {(r, θ) : 0 ≤ r ≤ f (θ)}. Further, assume that the curve is convex. Recall that a closed curve is convex if a line segment between any two of its points has no points exterior to the curve [1], [2, pp. 198-203]. We call these curves M-curves, because the curves are fundamental objects in Minkowski geometry, where they are called Minkowski circles or simply circles [3, 4]. That application is briefly discussed in the Section 4 but is not required for our purposes.Examples of M-curves are displayed in Figures 1 to 6. In order to express these curves as functions in rectangular coordinates, we need axes.

Upper bounds for minimum dilation triangulation in two special cases

Give a triangulation of a set of points on the plane, dilation of any two points is defined as the ratio between the length of the shortest path of these points and their Euclidean distance. Minimum dilation triangulation is a triangulation in which the maximum dilation between any pair of the points is minimized. We give upper bounds on the dilation of the minimum dilation triangulation for two kinds of point sets: An upper bound of nsin⁡(π/n)/2 for a centrally symmetric convex point set containing n points, and an upper bound of 1.19098 for a set of points on the boundary of a semicircle.

On Dimension-Free Variational Inequalities for Averaging Operators in $${\mathbb{R}^d}$$ R d

We study dimension-free L p inequalities for r-variations of the Hardy–Littlewood averaging operators defined over symmetric convex bodies in $${\mathbb{R}^d}$$ .

A GEOMETRIC APPROACH TO RADIAL CORRELATION TYPE PROBLEMS

A radial probability measure is a probability measure with a density (with respect to the Lebesgue measure) which depends only on the distances to the origin. Consider the Euclidean space enhanced with a radial probability measure. A correlation problem concerns showing whether the radial measure of the intersection of two symmetric convex bodies is greater than the product of the radial measures of the two convex bodies. A radial measure satisfying this property is said to satisfy the correlation property. A major question in this field is about the correlation property of the (standard) Gaussian measure. The main result in this paper is a theorem suggesting a sufficient condition for a radial measure to satisfy the correlation property. A consequence of the main theorem will be a proof of the correlation property of the Gaussian measure.

Efficient algorithms for discrepancy minimization in convex sets

A result of Spencer states that every collection of n sets over a universe of size n has a coloring of the ground set with of discrepancy . A geometric generalization of this result was given by Gluskin (see also Giannopoulos) who showed that every symmetric convex body with Gaussian measure at least , for a small , contains a point where a constant fraction of coordinates of y are in . This is often called a partial coloring result. While the proofs of both these results were inherently non‐algorithmic, recently Bansal (see also Lovett‐Meka) gave a polynomial time algorithm for Spencer's setting and Rothvoß gave a randomized polynomial time algorithm obtaining the same guarantee as the result of Gluskin and Giannopoulos. This paper contains several related results which combine techniques from convex geometry to analyze simple and efficient algorithms for discrepancy minimization. First, we prove another constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function. This implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer. Our second result suggests a new approach to obtain partial colorings, which is also valid for the non‐symmetric case. It shows that every (possibly non‐symmetric) convex body , with Gaussian measure at least , for a small , contains a point where a constant fraction of coordinates of y are in . Finally, we give a simple proof that shows that for any there exists a constant c &gt; 0 such that given a body K with , a uniformly random x from is in cK with constant probability. This gives an algorithmic version of a special case of the result of Banaszczyk.

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.

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

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.

Extremal Problems Related to Dual Gauss-John Position

In this paper, the extremal problem, min{( ΦK) :o ∈ ΦK Φ L, Φ ∈ GL(n)} , of two convex bodies K and L in is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min{((ΦK))p : ΦK Φ B2n, ΦK ∩ Sn-1 ≠ Φ, Φ ∈ GL(n)} . As an application, we give the geometric distance between the unit ball B2n and a centrally symmetric convex body K.

The Symplectic Size of a Randomly Rotated Convex Body

In this note we study the expected value of certain symplectic capacities of randomly rotated centrally symmetric convex bodies in the classical phase space.

Pairwise intersecting homothets of a convex body

We show that the maximum number of pairwise intersecting positive homothets of a d-dimensional centrally symmetric convex body, none of which contains the center of another in its interior, is at most 3d+1. Also, we improve upper bounds for cardinalities of k-distance sets in Minkowski spaces.