Universal cover in $\mathbb{ E}^{ n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in $\mathbb{ E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{ \tfrac{ n}{ 2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{ E}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{ E}^n$, $$ { \rm Vol}(U) \ge (1-o(1))^n{ \rm Vol}(J_n). $$
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