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

Now, for a given convex body $C$ and $\lambda$ ($\approx 1/2$), what is an upper bound for the number of homothetic copies of $C$ (with coefficient of homothety $\lambda$) required to cover $C$? And what other better bounds exist for some special convex objects?