Colorful and Quantitative Variations of Krasnosselsky's Theorem

Krasnosselsky’s art gallery theorem gives a combinatorial characterization of star-shaped sets in Euclidean spaces, similar to Helly’s characterization of finite families of convex sets with non-empty intersection. We study colorful and quantitative variations of Krasnosselsky’s result. In particular, we are interested in conditions on a set $K$ that guarantee there exists a measurably large set $K^{\mathrm{\prime }}$ such that every point in $K^{\mathrm{\prime }}$ can see every point in $K$. We prove results guaranteeing the existence of $K^{\mathrm{\prime }}$ with large volume or large diameter.