SCHWEITZER 2012 Problem 4

Let $K$ be a convex shape in the $n$ dimensional space, having unit volume. Let $S \subset K$ be a Lebesgue measurable set with measure at least $1-\varepsilon$ , where $0<\varepsilon<1/3$ . Prove that dilating $K$ from its centroid by the ratio of $2\varepsilon \ln(1/\varepsilon)$ , the shape obtained contains the centroid of $S$ .