MathDB

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

4