Embedding convex geometries and a bound on convex dimension

The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into $\mathbb{R}$ and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling.Comment: - Corrected attribution for Lemma 1 and Theorem 2 - Added an example related to generalized convex shellings of lower-bounded lattices and noted its relevance to convex dimension. - Added a section on embedding convex geometries as convex polygons, including a proof that any convex geometry may be embedded as convex polygons in R^2. - Extended the bibliography. Now 9 page

Asymptotic volume in Hilbert Geometries

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension

Isoperimetry of waists and local versus global asymptotic convex geometries

Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). The method relies on the new 'isoperimetry of waists' on the sphere due to Gromov

On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry $\Gamma = ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $\Gamma$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $\Gamma$. We will then investigate the transparency of Pl\"ucker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.Comment: 28 Pages/revised version after revie

Polarized non-abelian representations of slim near-polar spaces

In (Bull Belg Math Soc Simon Stevin 4:299-316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195-213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding