# Covering the convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~ \lambda c+x|c\in C$ } for some$x\in R^d$ is called homothetic copy of $C$. The number $\lambda>0$ is called as coefficient of homothety.

Now for given convex body $C$ and $\lambda (\approx 1/2)$, how many (upper bound) homothetic copies (with coffeicient of homothety $\lambda$) of $C$ would be require to cover the $C$? And what other better bounds exists for some special convex objects?

-

Any system of cubes with total volume $1$ can be arranged so as to cover a cube of volume $1/(2^d-1)$.