Is the following true?

CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ cannot be isometrically embedded in $\ C,\ $ for no $\ t>1.\ $ Then the center $c(C)$ of $C$ must belong to $B$, $\ c(C)\in B$.

Thank you Douglas Z. for pointing out the mess in my earlier formulation.

Sorry for a series of additional omissions. (Now the text is complete, I hope).

EDIT (after solutions of the original conjecture, by katz and Wlodek K)--As Wlodek Kuperberg has observed, an additional assumption about the n-dim volume (or area in 2-dim) of $B$ and $C$:

$$ |C|\ \le\ 2\cdot|B| $$

makes the conjecture true even without the earlier assumption about enlargement of $B$ (nor about symmetry).