The Orlicz centroid inequality for star bodies

Lutwak, Yang and Zhang established the Orlicz centroid inequality for convex bodies and conjectured that their inequality can be extended to star bodies. In this paper, we confirm this conjecture.     

The illumination conjecture for spindle convex bodies

The main principle of affine quantum gravity is the strict positivity of the matrix _boxclose_ab (x) \{ \hat g_{ab} (x)\} composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.     

椭球的Orlicz质心体(英文)

主要研究了Lutwak等所引入的Orlicz质心体(Lutwak E,Yang D,Zhang G.Orliczcentroid bodies.J.Differential Geom.,2010,84:365-387).利用Orlicz质心体在线性变换下的不变性,证明了椭球的Orlicz质心体仍是椭球.作为例子,计算了当取两个特定的凸函数时单位球的Orlicz质心体的支持函数.     

Determination of star bodies from p-centroid bodies

In this paper, we prove that an origin-symmetric star body is uniquely determined by its p-centroid body. Furthermore, using spherical harmonics, we establish a result for non-symmetric star bodies. As an application, we show that there is a unique member of $\Gamma_p\langle K\rangle$ characterized by having larger volume than any other member, for all real p?≥?1 that are not even natural numbers, where $\Gamma_p\langle K\rangle$ denotes the p-centroid equivalence class of the star body K.     

The Borsuk conjecture holds for convex bodies with a belt of regular points

Borsuk conjectured in 1933 that each bounded set inE ⁿ can be covered byn+1 sets of smaller diameter. It was disproved recently by Kahn and Kalai. However, Hadwiger proved earlier the conjecture for smooth convex bodies. We replace here the condition of smoothness by an assumption that the body has only a belt of regular points.Supported by a Canadian NSERC Grant.     

Some comments on floating and centroid bodies in the plane

Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \(K_w\) be the wet part of K. In this paper we study some properties of the locus of the centroid of \(K_w\) and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.     

The Lalonde-McDuff conjecture for nilmanifolds

We prove that any Hamiltonian bundle whose fiber is a nilmanifold c-splits.     

The Mordell-Lang conjecture for function fields

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin's homomorphism and of a certain analog in characteristic .

     

The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

The embedding conjecture for quasi-ordinary hypersurfaces

This paper has two objectives: we first generalize the theory of Abhyankar-Moh to quasi-ordinary polynomials, then we use the notion of approximate roots and that of generalized Newton polygons in order to prove the embedding conjecture for this class of polynomials. This conjecture-made by S.S. Abhyankar and A. Sathaye-says that if a hypersurface of the affine space is isomorphic to a coordinate, then it is equivalent to it.     

The Tate-Voloch conjecture for Drinfeld modules

We study the v-adic distance from the torsion of a Drinfeld module to an affine variety.     

The dimension conjecture for polydisc algebras

It is shown that form≠n the polydisc algebrasA(D ^m) andA(D ⁿ) are not isomorphic as Banach spaces. More precisely, there is no linear embedding of the dual spaceA(D ⁿ)* intoA(D ^m)* form<n. The invariant is infinite dimensional and is based on certain, multi-indexed martingales related to those considered by Davis et al. [10]. In the one-dimensional case, i.e. for the spaceA(D)*, a finite inequality is proved, implying thatA(D ²)* is not finitely representable inA(D)*. Extensions to algebras on products of strictly pseudoconvex domains are outlined. They imply in particular the non-isomorphism of certain algebras in the same number of variables, for instance A (D⁴) ≠ A (B₂xB₂).     

The center conjecture for non-exceptional buildings

We prove the center conjecture for spherical buildings of non-exceptional type. Our proof uses the point-line spaces associated with these buildings.     

The Baum-Connes conjecture for hyperbolic groups

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. Oblatum 20-VI-2001 & 24-VIII-2001?Published online: 15 April 2002 RID="*" ID="*"The second author is partially supported by NSF and MSRI.     

The gallai-younger conjecture for planar graphs

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG–X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.     

The composition conjecture for Abel equation

The composition conjecture for the Abel differential equation states that if all solutions in a neighborhood of the origin are periodic then the indefinite integrals of its coefficients are compositions of a periodic function. Several research articles were published in the last 20 years to prove the conjecture or a weaker version of it. The problem is related to the classical center problem of polynomial two-dimensional systems. The conjecture opens important relations with classical analysis and algebra. We give a widely accessible exposition of this conjecture and verify the conjecture for certain classes of coefficients.