Exact computation of the variational constant

Determine whether the variational quantity \(c_\alpha\), defined as the infimum of the differential-entropy functional over smooth compactly supported probability densities associated with the folded density \(f_{(\alpha)}\), can be computed exactly.

Background

The constant cαc_\alpha is introduced as an infimum involving the folded density f(α)f_{(\alpha)} and the one-dimensional projection densities f0,f1,f∞f_0,f_1,f_\infty. It is intended to capture the leading coefficient in the continuous-limit asymptotics of the sum-difference exponent.

The paper notes that two-dimensional Gaussian test functions perform reasonably well numerically but do not exactly extremize the functional, leaving the exact evaluation of cαc_\alpha unresolved.

References

It is not clear whether this quantity $c_\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional hf precisely.

hf:

h2(f(α))−max⁡(h(f0),h(f∞),h(fα))h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))

— Sum-difference exponents for boundedly many slopes, and rational complexity  (2511.15135 - Tao, 19 Nov 2025) in Section 1.2, immediately after Theorem 1.3 (“Continuous limit”)