Description
We discuss two Helly-type results with sharp Helly numbers. 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 present a Quantitative Helly-type theorem with the optimal Helly number 2d+1 for the pointwise minimum of logarithmically concave functions.
If time permits, we show 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. Joint work with Grigory Ivanov.