Exponential improvements for superball packing upper bounds

We prove that for all fixed $p>2$, the translative packing density of unit ${\mathrm{\ell }}_p$-balls in ${\mathbb{R}}^n$ is at most $2^{({\gamma }_p+o(1))n}$ with ${\gamma }_p<-1\mathrm{/}p$. This is the first exponential improvement in high dimensions since van der Corput and Schaake (1936).