Loomis-Whitney Inequality Research Articles

Loomis–Whitney inequalities on corank 1 Carnot groups

In this paper we provide another way to deduce the Loomis–Whitney inequality on higher dimensional Heisenberg groups \(\mathbb{H}^n\) based on the one on the first Heisenberg group \(\mathbb{H}^1\) and the known nonlinear Loomis–Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp–Lieb inequality and the subadditivity of the entropy developed in Carlen and Cordero-Erausquin (2009).

On the volume of the Minkowski sum of zonoids

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.

Low regularity well-posedness for KP-I equations: the dispersion-generalized case

We prove new well-posedness results for dispersion-generalized Kadomtsev–Petviashvili I equations in , which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show global well-posedness in . To this end, we combine resonance and transversality considerations with Strichartz estimates and a nonlinear Loomis–Whitney inequality. Moreover, we prove that for small dispersion, the equations cannot be solved via Picard iteration. In this case, we use an additional frequency dependent time localization.

Resonant decompositions and global well-posedness for 2D Zakharov-Kuznetsov equation in Sobolev spaces of negative indices

The Cauchy problem for Zakharov-Kuznetsov equation on R2 is shown to be global well-posed for the initial data in Hs provided s>−113. As conservation laws are invalid in Sobolev spaces below L2, we construct an almost conserved quantity by multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to the KdV equation, the main difficulty is to handle the resonant interactions due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.

Loomis–Whitney inequalities in Heisenberg groups

AbstractThis note concernsLoomis–Whitney inequalitiesin Heisenberg groups$$\mathbb {H}^n$$Hn:$$\begin{aligned} |K| \lesssim \prod _{j=1}^{2n}|\pi _j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb {H}^n. \end{aligned}$$|K|≲∏j=12n|πj(K)|n+1n(2n+1),K⊂Hn.Here$$\pi _{j}$$πj,$$j=1,\ldots ,2n$$j=1,…,2n, are thevertical Heisenberg projectionsto the hyperplanes$$\{x_j=0\}$${xj=0}, respectively, and$$|\cdot |$$|·|refers to a natural Haar measure on either$$\mathbb {H}^n$$Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group$$\mathbb {H}^1$$H1is a direct consequence of known$$L^p$$Lpimproving properties of the standard Radon transform in$$\mathbb {R}^2$$R2. In this note, we show how the Loomis–Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in$$\mathbb {H}^1$$H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound:$$\begin{aligned} \int _{\mathbb {H}^n} \prod _{j=1}^{2n} f_j(\pi _j(p))\;dp\lesssim \prod _{j=1}^{2n} \Vert f_j\Vert _{\frac{n(2n+1)}{n+1}} \end{aligned}$$∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1for all nonnegative measurable functions$$f_1,\ldots ,f_{2n}$$f1,…,f2non$$\mathbb {R}^{2n}$$R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in$$\mathbb {H}^n$$Hn, we mention the following sharper version of the classical geometric Sobolev inequality in$$\mathbb {H}^n$$Hn:$$\begin{aligned} \Vert u\Vert _{\frac{2n+2}{2n+1}} \lesssim \prod _{j=1}^{2n}\Vert X_ju\Vert ^{\frac{1}{2n}}, \qquad u \in BV(\mathbb {H}^n), \end{aligned}$$‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn),where$$X_j$$Xj,$$j=1,\ldots ,2n$$j=1,…,2n, are the standard horizontal vector fields in$$\mathbb {H}^n$$Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in$$\mathbb {H}^n$$Hn, where the Heisenberg vertical coordinate projections$$\pi _1,\ldots ,\pi _{2n}$$π1,…,π2nare replaced by more general families of mappings that allow us to apply the same inductive approach based on the$$L^{3/2}$$L3/2-$$L^3$$L3boundedness of an operator in the plane.

RECENT RESULTS ON CARDINALITY ESTIMATION AND INFORMATION THEORETIC INEQUALITIES

I would like to dedicate this little exposition to Prof. Phan Dinh Dieu, one of the giants and pioneers of Mathematics in Computer Science in Vietnam. In the past 15 years or so, new and exciting connections between fundamental problems in database theory and information theory have emerged. There are several angles one can take to describe this connection. This paper takes one such angle, influenced by the author's own bias and research results. In particular, we will describe how the cardinality estimation problem -- a corner-stone problem for query optimizers -- is deeply connected to information theoretic inequalities. Furthermore, we explain how inequalities can also be used to derive a couple of classic geometric inequalities such as the Loomis-Whitney inequality. A purpose of the article is to introduce the reader to these new connections, where theory and practice meet in a wonderful way. Another objective is to point the reader to a research area with many new open questions.

An Algebraic Brascamp\u2013Lieb Inequality

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions f_jin L^{q_j}(mathbb {R}^{n_j}), for j=1,ldots ,m, under some corresponding linear maps B_j. This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where B_j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Reverse Loomis-Whitney inequalities via isotropicity

Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tilde{\Lambda}(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tilde{\Lambda}(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tilde{\Lambda}(K)\leqslant(CL_K)^n$ 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 $(cL_K)^n\leqslant\tilde{\Lambda}(K)\leqslant(CL_K)^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.

On affine invariant and local Loomis–Whitney type inequalities

We prove various extensions of the Loomis–Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities, we estimate the constant in the Brascamp–Lieb inequality in terms of the vectors w i . Restricted and functional versions of the inequality will also be considered.

Inequalities about normalized Lp projection body

In this paper, we establish a Loomis-Whitney type inequality about volume normalized Lp projection body for p ≥ 1 with complete equality conditions for p ̸= 2. Meanwhile, an estimate for the weighted Lp zonoid is given.

Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

This paper is concerned with the Cauchy problem of the 2D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space Hs(R2) for s>−1/4, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in L2(R2).

A Grassmannian Loomis–Whitney inequality and its dual

A Grassmannian Loomis–Whitney inequality and its dual

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, 1q−homogeneous measures.

The Lp Gagliardo-Nirenberg-Zhang inequality

In the paper, the Lp version of Gagliardo-Nirenberg-Zhang inequality for isotropic measures is established. This inequality is the analytic analogue of the Lp Loomis-Whitney inequality and implies the classical Lp Sobolev inequality (without the best constant).

Computing from projections of random points

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call [Formula: see text]-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the [Formula: see text]-trivial sets. We characterize [Formula: see text]-bases as the sets computable from both halves of Chaitin’s [Formula: see text], and as the sets that obey the cost function [Formula: see text]. Generalizing these results yields a dense hierarchy of subideals in the [Formula: see text]-trivial degrees: For [Formula: see text], let [Formula: see text] be the collection of sets that are below any [Formula: see text] out of [Formula: see text] columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be [Formula: see text]. Furthermore, the corresponding cost function characterization reveals that [Formula: see text] is independent of the particular representation of the rational [Formula: see text], and that [Formula: see text] is properly contained in [Formula: see text] for rational numbers [Formula: see text]. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the [Formula: see text]-trivial sets; we can calculate from the family which [Formula: see text] it characterizes. We finish by studying the union of [Formula: see text] for [Formula: see text]; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic 81(3) (2016) 1028–1046], who showed that it is a proper subclass of the [Formula: see text]-trivial sets. We prove that all such sets are robustly computable from [Formula: see text], and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a [Formula: see text] subideal of the [Formula: see text]-trivial sets.

Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma

We prove a family of Sobolev inequalities of the form \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} \le C \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} where A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E) is a vector first-order homogeneous linear differential operator with constant coefficients, u is a vector field on \mathbb{R}^n and L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n}) is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo–Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn–Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis–Whitney inequality and Gagliardo's lemma.

On the Reverse Dual Loomis–Whitney Inequality

The dual Loomis–Whitney inequality provides the sharp lower bound for the volume of a convex body in terms of its $$(n-1)$$ -dimensional coordinate sections. In this paper, some reverse forms of the dual Loomis–Whitney inequality are obtained. In particular, we show that the best universal DLW-constant for origin-symmetric planar convex bodies is 1.

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.

Estimating volume and surface area of a convex body via its projections or sections

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 dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

Reverse Brascamp–Lieb inequality and the dual Bollobás–Thomason inequality

We prove that if $$f:{\mathbb {R}}^n\rightarrow [0,\infty )$$ is an integrable log-concave function with $$f(0)=1$$ and $$F_1,\ldots ,F_r$$ are linear subspaces of $${\mathbb {R}}^n$$ such that $$sI_n=\sum _{i=1}^rc_iP_i$$ where $$I_n$$ is the identity operator and $$P_i$$ is the orthogonal projection onto $$F_i$$ , then $$\begin{aligned} n^n\int \limits _{{\mathbb {R}}^n}f(y)^ndy\geqslant \prod _{i=1}^r\left( \,\int \limits _{F_i}f(x_i)dx_i\right) ^{c_i/s}. \end{aligned}$$ As an application we obtain the dual version of the Bollobas–Thomason inequality: if K is a convex body in $${\mathbb {R}}^n$$ with $$0\in \mathrm{int}(K)$$ and $$(\sigma _1,\ldots ,\sigma _r)$$ is an s-uniform cover of [n], then $$\begin{aligned} |K|^s\geqslant \frac{1}{(n!)^s}\prod _{i=1}^r|\sigma _i|!\prod _{i=1}^r|K\cap F_i|. \end{aligned}$$ This is a sharp generalization of Meyer’s dual Loomis–Whitney inequality.