Convexity of Omura’s strong converse exponent

Determine whether Omura’s strong converse exponent K_O(R) is convex as a function of the rate R for discrete memoryless channels, in contrast with the known convexity of the corresponding sphere-packing exponent below capacity.

Background

The paper studies the relationship between Omura’s exponent K_O(R) and the exact Dueck–Körner strong converse exponent K_DK(R) for rates above channel capacity. The authors explain that, below capacity, equality between the random-coding and sphere-packing exponents is governed by convexity and slope properties of the sphere-packing exponent. They note that an analogous structural argument is unavailable above capacity because the feasible set defining K_O(R), namely the set of channels V satisfying I(P,V) ≥ R, is not convex.

The convexity of K_O(R) would provide additional information about its growth and could clarify the geometry of the gap between Omura’s exponent and the exact strong converse exponent. The paper establishes the equality range between K_O(R) and K_DK(R) by other means, but does not resolve whether K_O(R) itself is convex.

References

No such structure is available above capacity: unlike its sphere packing counterpart, $K_{\mathrm{O}(R)$ is not known to be convex in $R$, as the constraint set ${V : I(P,V) \geq R}$ is not convex.

— On the Tightness of Omura's Strong Converse Exponent and Critical Rates Above Capacity  (2610.05791 - Joudeh et al., 5 Oct 2026) in Section 1, Introduction, paragraph beginning “Determining the full range of equality”