Origin-symmetric Bodies Research Articles

Optimal recovery and volume estimates

We study volumes of sections of convex origin-symmetric bodies in Rn induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes Wpγ, γ>0 in Lq on two-point homogeneous spaces in the difficult case, i.e. if 1<q≤p≤∞.

On the stability of the Lp-curvature

It is known that the Lp-curvature of a smooth, strictly convex body in Rn is constant only for origin-centered balls when 1≠p>−n, and only for balls when p=1. If p=−n, then the L−n-curvature is constant only for origin-symmetric ellipsoids. We prove ‘local’ and ‘global’ stability versions of these results. For p≥1, we prove a global stability result: if the Lp-curvature is almost a constant, then the volume symmetric difference of K˜ and a translate of the unit ball B is almost zero. Here K˜ is the dilation of K with the same volume as the unit ball. For 0≤p<1, we prove a similar result in the class of origin-symmetric bodies in the L2-distance. In addition, for −n<p<0, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with ‘almost’ constant Lp-curvature is ‘almost’ the unit ball. For p=−n, we prove a global stability result in R2 and a local stability result for n>2 in the Banach-Mazur distance.

Lattice Size and Generalized Basis Reduction in Dimension Three

The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in $$\mathbb R^2$$ based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of $$K=(P+(-P))^*$$ and to the lattice reduction with respect to the general norm that corresponds to K. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the “onion skins” algorithm) computes the lattice size of any convex body in $$\mathbb R^2$$ . We extend the work of Kaib and Schnorr to dimension three, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body $$K\subset \mathbb R^3$$ . We also explain how to recover the successive minima of K and the lattice size of P from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body $$P\subset \mathbb R^3$$ .

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

Origin-symmetric bodies of revolution with minimal Mahler volume in ℝ^3-a new proof

In (22), Meyer and Reisner proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R 3 , cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R 3 , 3-cubes have the minimal Mahler volume.

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball B d ⊂ R d is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to B d in Banach–Mazur distance converge to B d in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of B d . We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.

On a question of A. Koldobsky

We construct an example of a non-convex star-shaped origin-symmetric body D ⊂ R 3 such that its section function A D , ξ ( t ) : = area ( D ∩ { ξ ⊥ + t ξ } ) is decreasing in t ⩾ 0 for every fixed direction ξ ∈ S 2 .

Determination of convex bodies from Γ-section functions

In this paper, we prove that any polygon P in R 2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions among the polygons.

On strict inclusions in hierarchies of convex bodies

Let I k \mathcal I_k be the class of convex k k -intersection bodies in R n \mathbb {R}^n (in the sense of Koldobsky) and I k m \mathcal I_k^m be the class of convex origin-symmetric bodies all of whose m m -dimensional central sections are k k -intersection bodies. We show that 1) I k m ⊄ I k m + 1 \mathcal I_k^m\not \subset \mathcal I_k^{m+1} , k + 3 ≤ m &gt; n k+3\le m&gt;n , and 2) I l ⊄ I k \mathcal I_l \not \subset \mathcal I_k , 1 ≤ k &gt; l &gt; n − 3 1\le k&gt;l &gt; n-3 .

Modified Busemann-Petty problem on sections of convex bodies

The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ n with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes of central hyperplane sections so that the answer becomes affirmative in all dimensions.