  In 1957, Hadwiger made a conjecture that every $n$-dimensional convex body
can be covered by $2^n$ translations of its interior. The Hadwiger's covering
functional $\gamma_m(K)$ is the smallest positive number $r$ such that $K$ can
be covered by $m$ translations of $rK$. Due to Zong's program, we study the
Hadwiger's covering functional for the simplex and the cross-polytope. In this
paper, we give upper bounds for the Hadwiger's covering functional of the
simplex and the cross-polytope.

Metric Geometry