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Register a new userThe recent status of the volume product problem
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Authors
E. Makai Jr
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In this small survey we consider the volume product, and sketch some of the best upper and lower estimates known up to now, based on our paper [BMMR]. The author thanks the organizers of the conference in Jurata, March 2010, for their kind invitation, and the excellent atmosphere there. This paper is based on the talk of the author on that conference.
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We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds ${(M^n_i, p_i)}_{i=1}^{infty}$ with nonnegative Ricci curvature. Cheeger and Colding cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $ u$ on the limit space $(Y, p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y, p)$ and the volume growth of $ u$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if $lim_{r to infty} frac{ u(B_p(r))}{omega_n r^n} > alpha(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $alpha(k,n)$ are explicit and depend only on $n$, the dimension of the manifolds ${(M^n_i, p_i)}$, and $k$, the dimension of the homotopy in $(Y,p)$.
We prove that the largest convex shape that can be placed inside a given convex shape $Q subset mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when largest means largest volume and when it means largest surface area. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.
The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.
The Minkowski problem in Gaussian probability space is studied in this paper. In addition to providing an existence result on a Gaussian-volume-normalized version of this problem, the main goal of the current work is to provide uniqueness and existence results on the Gaussian Minkowski problem (with no normalization required).
After the initial discovery of the so-called spin crisis in the parton model in the 1980s, a large set of polarization data in deep inelastic lepton-nucleon scattering was collected at labs like SLAC, DESY and CERN. More recently, new high precision data at large x and in the resonance region have come from experiments at Jefferson Lab. These data, in combination with the earlier ones, allow us to study in detail the polarized parton densities, the Q^2 dependence of various moments of spin structure functions, the duality between deep inelastic and resonance data, and the nucleon structure in the valence quark region. Together with complementary data from HERMES, RHIC and COMPASS, we can put new limits on the flavor decomposition and the gluon contribution to the nucleon spin. In this report, we provide an overview of our present knowledge of the nucleon spin structure and give an outlook on future experiments. We focus in particular on the spin structure functions g_1 and g_2 of the nucleon and their moments.
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