Reverse Loomis-Whitney inequalities via isotropicity

Rogers-Shephard and local Loomis-Whitney type inequalities

We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers-Shephard type inequalities as well as some generalizations of the geometric Rogers-Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis-Whitney inequalities. We also obtain a sharp local Loomis-Whitney inequality.

On Extensions of the Loomis-Whitney Inequality and Ball's Inequality for Concave, Homogeneous Measures

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to Ball to the context of $q-$concave, $\frac{1}{q}-$homogeneous measures.

利阿科波罗斯广义双重卢米斯-怀特尼不等式中的等式情况通过巴瑟的反向布拉斯坎-利布不等式来刻画

Equality in Liakopoulos's generalized dual Loomis-Whitney inequality via Barthe's Reverse Brascamp-Lieb inequality

本文利用巴瑟反向布拉斯坎-利布不等式中的等式情况来刻画利阿科波罗斯体积估计中关于低维线性子空间截面的等式条件。

On the reverse Loomis-Whitney inequality

The present paper deals with the problem of computing (or at least estimating) the LW-number $λ(n)$, i.e., the supremum of all $γ$ such that for each convex body $K$ in $\mathbb{R}^n$ there exists an orthonormal basis $\{u_1,\ldots,u_n\}$ such that $$ vol_n(K)^{n-1} \geq γ\prod_{i=1}^n vol_{n-1} (K|u_i^{\perp}) , $$ where $K|u_i^{\perp}$ denotes the orthogonal projection of $K$ onto the hyperplane $u_i^{\perp}$ perpendicular to $u_i$. Any such inequality can be regarded as a reverse to the well-known classical Loomis--Whitney inequality. We present various results on such reverse Loomis--Whitn…

Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke …

Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality.