the optimization problems {#prelims}
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Let ${\mathcal{A}}_{LP}$ be the class of even continuous functions $f\in L^1({\mathbb{R}})$ satisfying the following conditions:

1.  ${\widehat}f(0)=f(0)=1$;

2.  ${\widehat}f\geq 0$;

3.  $f$ is eventually non-positive.

By eventually non-positive we mean that $f(x)\leq 0$ for all sufficiently large $|x|$. We then define the last sign change of $f$ by $$r(f)=\inf \big\{r>0: f(x)\leq 0 \text{ for } |x|\geq r\big\}.$$ It is easy to show that if $f\in A_{LP}$, then ${\widehat}f \in L^1({\mathbb{R}})$.

A remarkable breakthrough in the sphere problem was achieved by Cohn and Elkies in [@CE], where they showed that if $\Delta({\mathbb{R}}^d)$ is the highest sphere packing density in ${\mathbb{R}}^d$ then $$\Delta({\mathbb{R}}^d) \leq {\mathcal{Q}}(f)$$ for any $f\in  {\mathcal{A}}_{LP}({\mathbb{R}}^d)$ (this is the analogous class in higher dimensions defined for radial functions $f$), where $${\mathcal{Q}}(f) = \frac{\pi^{d/2}}{(d/2)!2^{d}}r(f)^d.$$ With this approach they generated numerical upper bounds, called linear programming bounds, for the packing density for dimensions up to $36$ (nowadays it goes much higher) that improved every single upper bound known at the time and still are the current best. These upper bounds in dimensions $8$ and $24$ revealed to be extremely close to the lower bounds given by the $E_8$ root lattice and the $\Lambda_{24}$ Leech lattice, revealing 