On two recent conjectures in convex geometry

Hilbert geometry for convex polygonal domains

We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.

关于高维凸体的Bezdek猜想及其对称中心

On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry

本文在高维空间中验证了Bezdek猜想，假设通过固定点的截面具有反射轴，且互补不变子空间平行于固定超平面，证明了在正交和仿射设置中成立。

The Viterbo's capacity conjectures for convex toric domains and the product of a $1$-unconditional convex body and its polar

In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex body $A\subset\mathbb{R}^{n}$ and its polar. We also give a direct calculus proof of the symmetric Mahler conjecture for $l_p$-balls.

On the X-ray number of almost smooth convex bodies and of convex bodies of constant width

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

球面凸几何中的二元运算

Binary Operations in Spherical Convex Geometry

该文研究球面凸几何中的二元运算，刻画了欧氏单位球面上凸体的二元运算性质，证明开半球内凸体对的非平凡投影协变运算本质上仅有凸包，且所有真球面凸体间连续投影协变二元运算必为平凡运算。

Spherical Localisation in Convex and Metric Geometry

Spherical localisation is a technique whose history goes back to M.Gromov and V.Milman. It's counterpart, the Euclidean localisation is extensively studied and has been put to great use in various branches of mathematics. The purpose of this paper is to quickly introduce spherical localisation, as well as demonstrate some of its applications in convex and metric geometry.

On the Hilbert Geometry of Convex Polytopes

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.

Conjectures and questions in convex geometry (of interest for quantum theory and other physical statistical theories)

Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.

The Soul Conjecture in Alexandrov Geometry in dimension 4

In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.

Another low-technology estimate in convex geometry

We give a short argument that for some C > 0, every n-dimensional Banach ball K admits a 256-round subquotient of dimension at least C n/(log n). This is a weak version of Milman's quotient of subspace theorem, which lacks the logarithmic factor.

仿射凸几何中的估值

Valuations in affine convex geometry

本文研究凸几何中满足保体积线性不变、连续且有限可加的估值构造，发现平移不变性下无新例，否则存在大量例子。

On the hyperplane conjecture for random convex sets

Let N > n, and denote by K the convex hull of N independent standard gaussian random vectors in an n-dimensional Euclidean space. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes.

关于Godbersen猜想的一个推广

On a generalization of Godbersen's conjecture

本研究提出Godbersen猜想的推广形式，将Schneider对Rogers-Shephard不等式的推广拓展至高阶差体，证明其对反块凸体成立，并建立了与高阶混合体积及Alesker乘积相关的等价表述。

Spherical geometry -- a survey on width and thickness of convex bodies

We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies …

On the Covering Densities of Quarter-Convex Disks

It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the non-symmetric case, we only know that the conjecture is true for triangles. In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals.

凸几何与离散几何中的Minkowski逐次极小

Minkowski's successive minima in convex and discrete geometry

本文综述了Minkowski逐次极小在凸几何与离散几何中的影响，探讨了其与格点计数器、内蕴体积等泛函的关系及相关猜想，并讨论了其在Siegel引理中的应用。

关于凸多面体在R^n中2n+4个法线猜想

On $2n+4$ normals conjecture for convex polytopes in $\mathbb{R}^n$

研究证明了在R^n中，每个通用简单多面体都存在一个点具有至少2n+4个法线，这一结果对应于光滑凸体法线问题的分段线性版本。

From Symplectic Measurements to the Mahler Conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

Bézout type inequality in convex geometry

We give a Bézout type inequality for mixed volumes, which holds true for any convex bodies. The key ingredient is the reverse Khovanskii-Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.

Spectrality of product domains and Fuglede's conjecture for convex polytopes

A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $Ω= A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result i…

Stability of the reverse Blaschke-Santalo inequality for unconditional convex bodies

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, w…

The Fuglede conjecture for convex domains is true in all dimensions

A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $Ω$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $Ω\subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ und…

凸集并集的体积估计与三维Kakeya集猜想

Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

本文研究三维空间中具有限制性管集合的并集体积，证明其体积接近最大，并由此推出所有Kakeya集的Minkowski和Hausdorff维数均为3。

The Log-Convex Density Conjecture and vertical surface area in warped products

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of $\mathbb{R}^{n}-\{0\}$ realized as the warped product $(0,\infty)\times_{r}S^{n-1}$, provi…

Confirmation of Matheron's conjecture on the covariogram of a planar convex body

The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…

On the Homothety Conjecture

Let $K$ be a convex body in $\bbR^n$ and $\d>0$. The homothety conjecture asks: Does $K_{\d}=c K$ imply that $K$ is an ellipsoid? Here $K_{\d}$ is the (convex) floating body and $c$ is a constant depending on $\d$ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\leq p\leq \infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\d} = c B^n_p$ if and only if $p=2$. We also show that the homothety conjecture is true for a general convex body $K$ if $\d$ is small enough. This improvs earlier results by Schütt and Werner …

Minimal volume product of convex bodies with certain discrete symmetries and its applications

We give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results of Mahler's conjecture and its non-symmetric version. In addition, we give partial answers for Viterbo's isoperimetric type conjecture in symplectic geometry from the view point of Mahler's conjecture.

Sumset estimates in convex geometry

Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic analogs of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and Lebesgue measure on ${\mathbb R}^n$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the …

Bezdek猜想的一个解

A solution to Bezdek's conjecture

本文证明在n≥2时，给定内径的λ-凸体中λ-凸透镜平均宽度最大，肯定回答Bezdek猜想；附加对称性时解决其任意内蕴体积的类似猜想；还证明固定外接半径的λ-凸体中λ-凸纺锤体平均宽度唯一最小。

normed平面中的蜂窝猜想及Dowker定理的一个α-凸变种

The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker

本文研究了normed平面中的蜂窝猜想，证明了正凸平铺的平均平方周长最小，并探讨了与Dowker定理α-凸变种相关的联系，进一步给出了关于平铺单元等周比的问题解答。

辛极性与马勒猜想

Symplectic polarity and Mahler's conjecture

本文提出辛自极凸体体积的猜想，并证明其等价于马勒猜想，同时给出了辛容量的上下界。

半空间中具有毛细边界的凸超曲面的 Alexandrov-Fenchel 不等式 II

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II

本文肯定回答了半空间凸毛细超曲面 Alexandrov-Fenchel 不等式的猜想，建立毛细凸体理论并证明一般混合体积不等式。

A New Excluding Condition towards the Soprunov-Zvavitch conjecture on Bezout-type inequalities

In 2015, I. Soprunov and A. Zvavitch have shown how to use the Bernstein-Khovanskii-Kushnirenko theorem to derive non-negativity of a certain bilinear form $F_Δ$, defined on (pairs of) convex bodies. Together with C. Saroglou, they proved non-negativity of $F_K$ characterizes simplices, among all polytopes. It is conjectured the characterization further holds among all convex bodies. Towards this conjecture, several necessary conditions on $K$ (for non-negativity of $F_K$), were derived. We give a new necessary condition, expressed with isoperimetric ratios, which provides a further step towar…

超形式、超流与凸几何

Superforms, supercurrents and convex geometry

本研究发展超形式演算作为凸几何工具，将其应用于凸体相关的赋值、不等式及方程，为凸几何研究提供新方法。

Integral Geometry about the visual angle of a convex set

In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.

Convexity of asymptotic geodesics in Hilbert Geometry

If $Ω$ is the interior of a convex polygon in $\mathbb{R}^{2}$ and $f,g$ two asymptotic geodesics, we show that the distance function $d\left(f\left(t\right),g\left(t\right)\right)$ is convex for $t$ sufficiently large. The same result is obtained in the case $\partial Ω$ is of class $C^{2}$ and the curvature of $\partial Ω$ at the point $f\left(\infty\right)=g\left(\infty\right) $ does not vanish. An example is provided for the necessity of the curvature assumption.

Tits Geometry, Arithmetic Groups and the Proof of a Conjecture of Siegel

We show that a locally symmetric space of noncompact type and with finite volume is quasi-isometric to the euclidean cone over a finite simplicial complex. A detailed analysis of metric properties yields a proof of a conjecture of Siegel.

A Generalized Central Limit Conjecture for Convex Bodies

The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by the Cheeger/Poincare/KLS constant. Here we propose a generalized CLT for marginals along random directions drawn from any isotropic log-concave distribution; namely, for $x,y$ drawn independently from isotropic log-concave densities $p,q$, the random variable $\langle x,y\rangle$ is close to Gaussian. Our main result is that this generalized CLT is quantita…

Viterbo's conjecture for certain Hamiltonians in classical mechanics

We study some particular cases of Viterbo's conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately. We manage to establish the conjecture for sublevel sets of convex $2$-homogeneous Hamiltonians of this kind in several particular cases. We also discuss open cases of this conjecture.

Isotropic constants and Mahler volumes

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.

维度Brunn-Minkowski猜想的加强意味着(B)-猜想

A strengthening of the dimensional Brunn-Minkowski conjecture implies the (B)-conjecture

本文证明了维度Brunn-Minkowski猜想的加强形式可推导出(B)-猜想，并通过遗传凸测度提供了该结论的替代证明。

On convexified packing and entropy duality

A 1972 duality conjecture due to Pietsch asserts that the entropy numbers of a compact operator acting between two Banach spaces and those of its adjoint are (in an appropriate sense) equivalent. This is equivalent to a dimension free inequality relating covering (or packing) numbers for convex bodies to those of their polars. The duality conjecture has been recently proved (see math.FA/0407236) in the central case when one of the Banach spaces is Hilbertian, which - in the geometric setting - corresponds to a duality result for symmetric convex bodies in Euclidean spaces. In the present paper…

Higher Lelong numbers and convex geometry

We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh-Khovanskii, Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented) Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly's conject…

有序超域上的凸几何

Convex geometry over ordered hyperfields

本文开创有序超域上的凸几何研究，定义凸集与半空间，证明Helly、Radon和Carathéodory定理的超域类比，并分类半空间分离成立及严格有序超域。

Coxeter group in Hilbert geometry

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $Γ$, find the maximal $Γ$-invariant convex, when there is a unique $Γ$-invariant convex, when the convex $Ω$ is strictly convex, when we can find a $Γ$-…

Subdifferentials and Minimizing Sard Conjecture in Sub-Riemannian Geometry

We use techniques from nonsmooth analysis and geometric measure theory to provide new examples of complete sub-Riemannian structures satisfying the Minimizing Sard conjecture. In particular, we show that complete sub-Riemannian structures associated with distributions of co-rank 2 or generic distributions of rank $\ge$ 2 satisfy the Minimizing Sard conjecture.

Dvoretzky type theorems for multivariate polynomials and sections of convex bodies

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even…

Approximability of convex bodies and volume entropy in Hilbert geometry

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).

Viterbo's conjecture for Lagrangian products in $\mathbb{R}^4$ and symplectomorphisms to the Euclidean ball

We use the generalized Minkowski billiard characterization of the EHZ-capacity of Lagrangian products in order to reprove that the $4$-dimensional Viterbo conjecture holds for the Lagrangian products (any triangle/parallelogram in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) and extend this fact to the Lagrangian products (any trapezoid in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$). Based on this analysis, we classify equality cases of this version of Viterbo's conjecture and prove that most of them can be proven to be symplectomorphic to Euclidean balls. As a by-…

Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.