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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.
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. Ch
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 m
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 pr
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 existen
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