Shapes of polyhedra, mixed volumes and hyperbolic geometry

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020)

Modulation of glycolysis induction in primary cultures of rabbit kidney proximal tubule cells: the role of shaking, glucose and insulin

We have assessed the impact of increasing oxygen availability on cellular phenotype expression of rabbit proximal tubule cells in primary culture developed with variable glucose and/or insulin contents. To mitigate hypoxia at the cell/medium interface, cells were shaken for the whole culture duration and their expressed phenotype was compared with those expressed by static cultures. O2 and CO2 tensions were kept constant in the incubator atmosphere. Glycolysis and gluconeogenesis pathways, detoxication system, and mitochondrial, apical and basolateral membrane marker enzyme activities were assessed. This study showed that the induction of glycolysis which appear in primary cultures of proximal tubule cells may be partially prevented by continuously shaking the cultures. This effect was more marked in the presence of glucose, suggesting better substrate oxidation in shaken cultures