Helly numbers for Quantitative Helly-type results

Abstract: We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number (2d) concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number (2d+1) for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes) (3d+1); however, we have no reason to believe that this bound is sharp.