The hyperbolic linear programming bound for sphere packing density

Let $f \colon [0,\infty) \to \mathbb{R}$ be continuous and integrable on $\mathbb{H}^n$, and suppose

$$f(x) \le 0 \quad\text{for all } x \ge 2r,$$

while $\widehat{f}(\lambda) \ge 0$ for all $\lambda>0$ and $\widehat{f}(0)>0$. Here $\Delta_{\mathbb{H}^n}(r)$ denotes the maximal density of a packing of radius-$r$ balls in hyperbolic $n$-space, $\widehat{f}$ is the radial Fourier transform, and $\operatorname{vol}(B_r^n)$ is the volume of a radius-$r$ ball. Hyperbolic linear programming bound. Under these hypotheses,

$$\Delta_{\mathbb{H}^n}(r) \le \operatorname{vol}(B_r^n)\frac{f(0)}{\widehat{f}(0)}.$$

This is the hyperbolic analogue of the Cohn–Elkies linear programming method for Euclidean sphere packing. The authors state it as a conjecture because they do not know whether the stronger assumption $\widehat{f}(\lambda)\ge0$ everywhere, rather than only on the Plancherel support, is truly necessary; that stronger assumption is used in the proof of their subsequent theorem.