We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in R3 and we establish an improved inequality in R2. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the Lp-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case p=2, in any dimension.

Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\mathcal P}(K)$, which is still open. We also show connections of ${\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. We observe that these ideas give a large family of convex sets in every dimension for which the Mahler conjecture holds true. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a {\mf new} stability result.

Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.

Let f be an integrable log- concave function on R-n with the center of mass at the origin. We show that integral(infinity)(0) f(s0) ds >= e(-n) integral(infinity)(-infinity) f(s0) ds for every 0 is an element of Sn-1, and the constant e(-n) is the best possible.

Let $K$ be a convex body in $\mathbb R^n$. We prove that in small codimensions, the sections of a convex body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies.

Let $f$ be an integrable log-concave function on ${\mathbb R^n}$ with the center of mass at the origin. We show that $\int\limits_0^{\infty}f(s\theta)ds\ge e^{-n}\int\limits_{-\infty}^{\infty}f(s\theta)ds$ for every $ \theta\in S^{n-1}$, and the constant $e^{-n}$ is the best possible.

Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endowed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local maximizer for the isotropy constant $L_K$.

High molecular weight polysilsesquioxane lithium salts were prepared. Ion self-diffusion coefficients and transference numbers in acetonitrile solutions were quantified by PFG NMR spectroscopy.

We answer in the negative a question by Grünbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Grünbaum about the “size” of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of ℝ n , is dense in the set of convex bodies. Crucial to establish these results are new affine invariant points, not previously considered in the literature.

An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^{p(K)})=p(K) for every convex body K, where K^{p(K)} denotes the polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.

If phi : R-n -> R boolean OR { + infinity} is a convex function such that 0 < integral e(-phi(x))dx < + infinity, the Mahler product of phi is defined by [GRAPHICS] where L-z phi (y) = sup(x is an element of Rn) < x - z, y - z > - phi(x) is the Legendre transform of phi with respect to z. We prove that on the set of all convex functions (convex even functions) the Mahler product has no local minimum at any function (even function) having some regular point. A similar result was obtained in [RSW] for convex bodies.

A straightforward and versatile sol–gel process for the design and preparation of single-ion conductor flexible membranes.

Hybrid organic–inorganic proton conducting membranes based on arylsulfonic acid (Ar-SO3H) groups and poly(ethylene glycol) (PEG) units were easily obtained in one step synthesis by co-hydrolysis and polycondensation of 2-(4-Chlorosulfonylphenyl)ethyl trimethoxysilane and bis-silylated polyethylene oxide. The hydrolysis of SO2Cl groups into SO3H was achieved during the sol–gel process by the in situ generated HCl acting as catalyst for the hydrolysis and polycondensation reactions. All membranes were characterized by 1H, 13C and 29Si MAS solid-state NMR, elementary analysis, Scanning Electron Microscopy (SEM) and X-ray photoelectron (XPS) spectroscopy. Their thermal and thermomechanical properties were investigated by thermogravimetric analysis (TGA), differential scanning calorimetry (DSC) and dynamic mechanical analysis (DMA), respectively.. The thermomechanical and conductive properties were found to be dependent on the PEG chain length and the PMMA amount. The proton conductivities of these materials were found to be approximately 10−2Scm−1 at 20°C.

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(K − K)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

This paper is mainly concerned with the structure of the centre of a vector lattice. A special attention is paid in the case when E=Lp, p≥1. In this paper we give some characterizations of dense vector sublattices of the centre. Those characterizations will be applied in several directions. At the end of this work we compare various fullness and richness properties of the centre of a vector lattice.

We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product $\mathcal{P}(K)=\min_{z\in\operatorname{int}(K)}|K||K^{z}|$ , where K⊂ℝ n is a convex body and K z ={y∈ℝ n :(y−z)⋅(x−z)≤1 for all x∈K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K⊂ℝ3 is the convex hull of two 2-dimensional convex bodies, then $\mathcal{P}(K) \geq \mathcal{P}(\varDelta ^{3})$ , where Δ 3 is a 3-dimensional simplex, thus confirming the 3-dimensional case of Mahler conjecture, for this class of bodies. A similar result is provided for the symmetric case, where we prove that if K⊂ℝ3 is symmetric and the convex hull of two 2-dimensional convex bodies, then $\mathcal{P}(K) \geq \mathcal{P}(B_{\infty}^{3})$ , where $B_{\infty}^{3}$ is the unit cube.

We present a method that allows us to prove that the volume product of polygons in ℝ 2 with at most n vertices is bounded from above by the volume product of regular polygons with n vertices. The same method shows that the volume product of polygons is bounded from below by the volume product of triangles (or parallelograms in the centrally symmetric case). These last results give a new proof of theorems of K. Mahler. The cases of equality are completely described.

Abstract Let L be a convex body in n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI(L, z). This generalizes the classical intersection bodyI(L, z) (whose radial function at u ∈ Sn−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.