Sharp area–shortest closed geodesic inequality for convex surfaces in R^3

Establish the sharp (optimal) inequality relating the area of a convex surface M^2 ⊂ R^3 to the length Λ(M) of its shortest closed geodesic. In particular, determine the optimal constant or functional form giving the best lower bound for Area(M) in terms of Λ(M).

Background

A fundamental question in Riemannian geometry asks for lower bounds on volume (or area) in terms of geodesic length scales. For S2, Croke proved a lower bound in terms of the shortest closed geodesic; for higher‑dimensional spheres this remains open. For convex hypersurfaces in all dimensions, Treibergs and Croke established nonsharp bounds.

Despite these advances, the precise optimal relation between area and the shortest closed geodesic length is unresolved even in the classical setting of convex surfaces in R3. Identifying the exact constant (or optimal inequality) would close a longstanding gap between known nonsharp bounds and conjectural sharp behavior.

References

This is known to be true for the $2$-sphere by work of Croke , open in general for the sphere of dimension $n \geq 3$, and true for convex hypersurfaces in all dimensions by work of Treibergs and Croke ; however, the sharp inequality between area and the length of the shortest closed geodesic is not known even for convex surfaces $M2$ in $R3$.

— Area and antipodal distance in convex hypersurfaces  (2604.02667 - Dibble et al., 3 Apr 2026) in Introduction (Section 1)