Covering convex solids by greater homotheties

Let $K$ be a convex solid of Euclidean space $E^n$, with $\operatorname{bd}K$ and $\operatorname{int}K$ being its boundary and interior. The paper solves the problem of the possibility of covering $K$ by sets homothetic to $\operatorname{int}K$, with the ratio of the homotheties being greater than unity and the centers being in $E^n\setminus\operatorname{int}K$, while, should such a covering exist, an estimate is provided of the least cardinality of the family of sets covering $K$.