On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

TGFβ1 expression in colonic mucosa: modulation by dietary lipids

Transforming growth factor beta1 (TGFβ1) is fundamental to maintain the intestinal epithelial cell homeostasis through its control action on cell proliferation, differentiation and apoptosis. TGFβ1 dysregulation has been observed in several chronic human diseases, including ulcerative colitis, Crohn’s disease and colon carcinoma. In the first two conditions, a marked oxidative stress is consistently present, while in the third one, levels of reactive oxygen species tend to be significantly lower than in the surrounding normal tissue. Lipid-derived compounds such as the aldehyde 4-hydroxynonenal (HNE) or cholesterol oxidation products (oxysterols) were shown able to induce expression and synthesis of TGFβ1, an event which can be detrimental or beneficial, essentially depending on its actual intensity. Understanding how specific dietary lipids may influence the complex molecular signaling underlying this cytokine expression, may provide new indications for therapeutic and preventive strategies in inflammatory bowel diseases and colon carcinoma