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on A Linear Programming Bound (AlphaEvolve problem 13)

The statement

For any dimension $n$ , let $C(n)$ denote the quantity

C(n) \coloneqq \frac{\pi^{n/2}}{\Gamma(n/2+ 1)} \inf_{f} \frac{(r/2)^n f(0)}{\hat f(0)}

where $f$ ranges over integrable continuous functions $f \coloneqq \mathbb{R}^n \to \mathbb{R}$ , not identically zero, with $\hat f(\xi) \geq 0$ for all $\xi$ and $f(x) \leq 0$ for all $|x| \geq r$ for some $r>0$ . Establish upper bounds for $C(n)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.