Origin-symmetric Convex Bodies Research Articles

Thick sections of convex bodies

Let P and Q be origin-symmetric convex bodies in Rn such that for every slab (thick section) of a fixed width 2t, symmetric about the origin, its intersections with P and Q have equal volumes. Is then necessarily P=Q? We show that this is true in the class of origin-symmetric convex polytopes. We also study a modified version of this problem, when one of the bodies is a Euclidean ball and both bodies have the same volume. Finally we discuss a related problem concerning sections of convex bodies by hyperplanes that are distance t from the origin.

Isomorphic Busemann–Petty problem for sections of proportional dimensions

The main result of this note is a solution to the isomorphic Busemann–Petty problem for sections of proportional dimensions, as follows. Suppose that 0<λ<1, k>λn, and K,L are origin-symmetric convex bodies in Rn satisfying the inequalities|K∩H|≤|L∩H|,∀H∈Grn−k, where Grn−k is the Grassmanian of (n−k)-dimensional subspaces of Rn, and |K| stands for volume of proper dimension. Then|K|n−kn≤Ck((1−log⁡λ)3λ)k|L|n−kn, where C is an absolute constant.

The planar Busemann-Petty centroid inequality and its stability

In Centro-affine invariants for smooth convex bodies [Int. Math. Res. Notices. DOI 10.1093/imrn/rnr110, 2012] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty .$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$, in the class of smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow evolves appropriately normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack estimate for this flow, we prove a stability version of the planar Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we prove that the convergence of normalized solutions in the Hausdorff metric can be improved to convergence in the $\mathcal {C}^{\infty }$ topology.

Slicing inequalities for measures of convex bodies

We consider the following problem. Does there exist an absolute constant C so that for every n∈N, every integer 1≤k<n, every origin-symmetric convex body L in Rn, and every measure μ with non-negative even continuous density in Rn,(1)μ(L)≤CkmaxH∈Grn−k⁡μ(L∩H)|L|k/n, where Grn−k is the Grassmanian of (n−k)-dimensional subspaces of Rn, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the slicing problem, a major open problem in convex geometry.It was proved in [18,19] that (1) holds for arbitrary origin-symmetric convex bodies, all k and all μ with C≤O(n). In this article, we prove inequality (1) with an absolute constant C for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove that for every λ∈(0,1) there exists a constant C=C(λ) so that inequality (1) holds for every n∈N, every origin-symmetric convex body L in Rn, every measure μ with continuous density and the codimension of sections k≥λn. The proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies. In the last section, we show that for some measures the behavior of minimal sections may be very different from the case of volume.

An isomorphic version of the Busemann–Petty problem for arbitrary measures

The Busemann–Petty problem for an arbitrary measure $$\mu $$ with non-negative even continuous density in $${\mathbb R}^n$$ asks whether origin-symmetric convex bodies in $${\mathbb R}^n$$ with smaller $$(n-1)$$ -dimensional measure $$\mu $$ of all central hyperplane sections necessarily have smaller measure $$\mu .$$ It was shown in Zvavitch (Math Ann 331:867–887, 2005) that the answer to this problem is affirmative for $$n\le 4$$ and negative for $$n\ge 5$$ . In this paper we prove an isomorphic version of this result. Namely, if $$K,M$$ are origin-symmetric convex bodies in $${\mathbb R}^n$$ such that $$\mu (K\cap \xi ^\bot )\le \mu (M\cap \xi ^\bot )$$ for every $$\xi \in {\mathbb S}^{n-1},$$ then $$\mu (K)\le \sqrt{n} \mu (M).$$ Here $$\xi ^\bot $$ is the central hyperplane perpendicular to $$\xi .$$ We also study the above question with additional assumptions on the body $$K$$ and present the complex version of the problem. In the special case where the measure $$\mu $$ is convex we show that $$\sqrt{n}$$ can be replaced by $$cL_n,$$ where $$L_n$$ is the maximal isotropic constant. Note that, by a recent result of Klartag, $$L_n \le O(n^{1/4})$$ . Finally we prove a slicing inequality $$\begin{aligned} \mu (K)\le C n^{1/4} \max _{\xi \in {\mathbb S}^{n-1}} \mu (K \cap \xi ^\perp ) \mathrm{vol}_{n}(K)^{\frac{1}{n}} \end{aligned}$$ for any convex even measure $$\mu $$ and any symmetric convex body $$K$$ in $${\mathbb R}^n,$$ where $$C$$ is an absolute constant. This inequality was recently proved in Koldobsky (Adv Math 254:33–40, 2014) for arbitrary measures with continuous density, but with $$\sqrt{n}$$ in place of $$n^{1/4}.$$

The 3-ball is a local pessimum for packing

It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls. We also show that in dimensions 4, 5, 6, 7, 8, and 24 there are origin-symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be.

A new proof of the Log-Brunn–Minkowski inequality

Recently Boroczky, Lutwak, Yang and Zhang have proved the log-Brunn–Minkowski inequality which is stronger than the classical Brunn–Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper presents a new proof of this inequality and proves the uniqueness of the cone-volume measure by using the log-Minkowski inequality.

On the Mean-Width of Isotropic Convex Bodies and their Associated Lp-Centroid Bodies

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of $K$, and $C>0$ is a universal constant. This improves the previous best-known estimate $M^*(K) \leq C n^{3/4} L_K$. Up to the power of the $\log(n)$ term and the $L_K$ one, the improved bound is best possible, and implies that the isotropic position is (up to the $L_K$ term) an almost $2$-regular $M$-position. The bound extends to any arbitrary position, depending on a certain weighted average of the eigenvalues of the covariance matrix. Furthermore, the bound applies to the mean-width of $L_p$-centroid bodies, extending a sharp upper bound of Paouris for $1 \leq p \leq \sqrt{n}$ to an almost-sharp bound for an arbitrary $p \geq \sqrt{n}$. The question of whether it is possible to remove the $L_K$ term from the new bound is essentially equivalent to the Slicing Problem, to within logarithmic factors in $n$.

ORIGIN-SYMMETRIC CONVEX BODIES WITH MINIMAL MAHLER VOLUME IN ℝ2

In this paper, a new proof of the following result is given: The product of the volumes of an origin-symmetric convex bodies K in <TEX>$\mathbb{R}^2$</TEX> and of its polar body is minimal if and only if K is a parallelogram.

A characterization of Blaschke addition

A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of Lévy–Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.

A [formula omitted] estimate for measures of hyperplane sections of convex bodies

The hyperplane (or slicing) problem asks whether there exists an absolute constant C so that for any origin-symmetric convex body K in Rn|K|n−1n⩽Cmaxξ∈Sn−1|K∩ξ⊥|, where ξ⊥ is the central hyperplane in Rn perpendicular to ξ, and |K| stands for volume of proper dimension. The problem is still open, with the best-to-date estimate C∼n1/4 established by Klartag, who slightly improved the previous estimate of Bourgain. It is much easier to get a weaker estimate with C=n.In this note we show that the n estimate holds for arbitrary measure in place of volume. Namely, if L is an origin-symmetric convex body in Rn and μ is a measure with non-negative even continuous density on L, thenμ(L)⩽nnn−1cnmaxξ∈Sn−1μ(L∩ξ⊥)|L|1/n, where cn=|B2n|n−1n/|B2n−1|<1, and B2n is the unit Euclidean ball in Rn. We deduce this inequality from a stability result for intersection bodies.

The lower dimensional Busemann–Petty problem in the complex hyperbolic space

The lower dimensional Busemann–Petty problem asks whether origin-symmetric convex bodies in Rn with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for k=2,3. We study this problem in the complex hyperbolic n-space HCn and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.

Volume inequalities for L 0 -Minkowski combination of convex bodies

Recently Boroczky, Lutwak, Yang and Zhang proved the -Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper extends their results to m ( ) origin-symmetric convex bodies in the plane. Moreover, relying on the recent results of Schuster and Weberndorfer, volume inequalities for -Minkowski combination of origin-symmetric convex bodies in and its dual form are established in this paper. MSC:52A20, 52A40.

On the Stability of the p-Affine Isoperimetric Inequality

Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in ℝ2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in ℝ2 such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.

The Busemann–Petty problem in the complex hyperbolic space

AbstractThe Busemann–Petty problem asks whether origin-symmetric convex bodies in ℝn with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 4 and negative if n ≥ 5. We study this problem in the complex hyperbolic n-space ℍnℂ and prove that the answer is affirmative for n ≤ 2 and negative for n ≥ 3.

The Busemann theorem for complex p-convex bodies

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.

The log-Brunn–Minkowski inequality

For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID

For <TEX>$0</TEX><TEX><</TEX><TEX>p{\leq}{\infty}$</TEX> and a convex body <TEX>$K$</TEX> in <TEX>$\mathbb{R}^n$</TEX>, Lutwak, Yang and Zhang defined the concept of dual <TEX>$L_p$</TEX>-centroid body <TEX>${\Gamma}_{-p}K$</TEX> and <TEX>$L_p$</TEX>-John ellipsoid <TEX>$E_pK$</TEX>. In this paper, we prove the following two results: (i) For any origin-symmetric convex body <TEX>$K$</TEX>, there exist an ellipsoid <TEX>$E$</TEX> and a parallelotope <TEX>$P$</TEX> such that for <TEX>$1{\leq}p{\leq}2$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>$E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$</TEX> and <TEX>$V(E)=V(K)=V(P)$</TEX>; For <TEX>$2{\leq}p{\leq}{\infty}$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>$2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$</TEX> and <TEX>$V(E)=V(K)=V(P)$</TEX>. (ii) For any convex body <TEX>$K$</TEX> whose John point is at the origin, there exists a simplex <TEX>$T$</TEX> such that for <TEX>$1{\leq}p{\leq}{\infty}$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>${\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT$</TEX> and <TEX>$V(K)=V(T)$</TEX>.

New thoughts on Weinberger’s first and second integral bounds for Green’s function

New thoughts about the first and second integral bounds of Hans F. Weinberger for Green’s functions of uniformly elliptic equations are presented by extending the bounds to two optimal monotone principles, but also further explored via: (i) discovering two new sharp Green-function- involved isoperimetric inequalities; (ii) verifying the lower dimensional Polya conjecture for the lowest eigenvalue of the Laplacian; (iii) sharpening an eccentricity-based lower bound for the Mahler volumes of the origin-symmetric convex bodies.

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.