[Paper Review] A hyperplane inequality for measures of unconditional convex bodies

Alexander Koldobsky|arXiv (Cornell University)|Dec 26, 2013

Point processes and geometric inequalities1 references3 citations

TL;DR

This paper establishes a hyperplane inequality for measures of unconditional convex bodies, proving that the measure of such a body is bounded by a universal constant times the maximum measure of its central hyperplane sections multiplied by the body's volume to the power 1/n. The result extends Bourgain's volume inequality to arbitrary even continuous measures using stability properties of intersection bodies and linear transformations of lp-balls.

ABSTRACT

We prove an inequality that extends to arbitrary measures the hyperplane inequality for volume of unconditional convex bodies originally observed by Bourgain.

Motivation & Objective

• To extend the hyperplane inequality from volume to arbitrary measures for unconditional convex bodies.
• To establish a uniform bound involving the maximum measure of central hyperplane sections and the volume of the body.
• To generalize the hyperplane inequality to duals of convex bodies with bounded volume ratio.
• To leverage stability results for intersection bodies to derive measure-based inequalities.
• To provide a unified framework for measure comparison in symmetric convex geometry using functional and geometric tools.

Proposed method

• Use of Lozanovskii's theorem to embed an unconditional convex body L into a scaled L1-ball and its dual, enabling the use of intersection body properties.
• Construction of a linear operator T such that T(B∞ⁿ) ⊂ L ⊂ nT(B1ⁿ), linking L to an intersection body K.
• Definition of a density function f = χK + gχL, where g is the density of the measure μ, ensuring f ≥ 1 on K.
• Application of a stability inequality for intersection bodies (Proposition 1) to bound the integral of f over K.
• Use of volume ratios and polar bodies to relate the measure of L to that of its dual, particularly through ellipsoids.
• Estimation of |K|^{1/n} in terms of |L|^{1/n} using volume formulas for ℓp-balls and determinant bounds.

Experimental results

Research questions

• RQ1 Can the hyperplane inequality for volume of unconditional convex bodies be extended to arbitrary measures with even continuous densities?
• RQ2 What is the optimal universal constant C in the measure-based hyperplane inequality for unconditional bodies?
• RQ3 How does the volume ratio of the polar body influence the measure inequality for duals of bodies with bounded volume ratio?
• RQ4 To what extent can stability results for intersection bodies be used to derive measure comparison inequalities?
• RQ5 Is the dependence on the volume ratio in the dual case tight or can it be improved?

Key findings

• There exists an absolute constant C such that for any unconditional convex body L and any measure μ with even continuous non-negative density, μ(L) ≤ C × maxξ∈Sⁿ⁻¹ μ(L ∩ ξ⊥) × |L|^{1/n}.
• The constant C can be taken as e (approximately 2.718), and asymptotically tends to about 0.5√e as n → ∞.
• The inequality is proven via a stability result for intersection bodies, where the body K = nT(B₁ⁿ) is shown to be an intersection body.
• For duals of bodies with bounded volume ratio, the inequality includes a factor of the volume ratio of the polar body, with C depending only on absolute constants.
• The result generalizes previous inequalities for intersection bodies and k-intersection bodies, with the best-known constant C = n/(n−1)cn for intersection bodies.
• The method applies to both unconditional bodies and duals of bodies with bounded volume ratio, unifying two classes under a single measure-based hyperplane inequality.