Johannes Hosle
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.
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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。
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…
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 …
Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tildeΛ(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tildeΛ(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tildeΛ(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tildeΛ(K)\l…
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.
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.
We prove that if $P,\mathcal{L}$ are finite sets of $δ$-separated points and lines in $\mathbb{R}^{2}$, the number of $δ$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\mathcal{L}|^{2/3} \cdot δ^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$ |K| \lesssim |π_{x}(K)|^{2/3} \cdot |π_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}. $$ Here $π_{x}$ and $π_{y}$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\cdot|$ refers to natural Haar measure on either $\…
The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a 'box'. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the Loomis-Whitney inequality to obtain a stability result for the edg…
The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least "outer volume ratio distance" from the class of intersection bodies of projections of at least proportional dimen…
We verify the Rota-Heron-Welsh conjecture for matroids realizable as c-arrangements: the coefficients of the characteristic polynomial of the associated matroid are log-concave. This family of matroids strictly contains that of complex hyperplane arrangements. Our proof combines the study of intrinsic volumes of certain extensions of arrangements and the Levy--Milman measure concentration phenomenon on realization spaces of arrangements.
On the volume of the Minkowski sum of zonoids
本文研究zonoid的Minkowski和体积相关不等式，证明多个局部几何不等式与体积对数次模性及Dembo-Cover-Thomas猜想等价，并在三维、二维及$L_2$理论中验证或加强结论。
The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke-Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's c…
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