In 1957, Hadwiger conjectured that every n-dimensional convex body can be covered by \(2^n\) translations of its interior. The covering functional f(K), defined as the smallest positive number r such that K can be covered by \(2^n\) translations of rK, provides a natural framework for studying this conjecture. In particular, Hadwiger’s conjecture is equivalent to the statement that \(\max _{K}f(K)\le c<1\) for some positive constant c. In this work, we investigate the covering functionals for two fundamental classes of convex bodies: the n-dimensional simplex \(S_n\) and the cross-polytope \(C_n^\star \) . We establish the upper bounds: \(\begin{aligned} \underset{n\rightarrow \infty }{\text {lim sup}}f(S_n)\le 0.773\cdots ~~~~\text {and}~~~~ \underset{n\rightarrow \infty }{\text {lim sup}}f(C_n^\star )\le 0.824\cdots . \end{aligned}\) Moreover, the method developed in this study can be further extended to investigate the covering functionals of quarter- \(\ell _p\) balls and \(\ell _p\) balls. These results establish the first non-trivial asymptotic upper bounds for the covering functionals of these fundamental convex bodies. Furthermore, they imply estimates for the dyadic entropy numbers of identity operators on finite-dimensional \(\ell _p^n\) spaces.
