Alexandrov’s conjecture on maximal area at fixed intrinsic diameter

Prove Alexandrov’s conjecture that among all convex surfaces M^2 in Euclidean 3‑space R^3 with fixed intrinsic diameter d, the unique surface of maximal area is the doubled disk of radius d/2.

Background

The conjecture identifies the doubled disk (two congruent flat disks glued along their boundary) as the extremizer of area under a fixed intrinsic diameter constraint among convex surfaces in R3. It has motivated work on diameter–area extremal problems and is linked to intrinsic–extrinsic comparisons for convex bodies.

The present paper proves lower bounds connecting area to displacement under maps on convex hypersurfaces; sharpening intrinsic–extrinsic inequalities could yield converse statements related to Alexandrov’s conjecture.

References

A long-standing conjecture of Alexandrov states that, among all convex surfaces $M2$ in $R3$ with intrinsic diameter $d$, the unique surface of maximal area is the doubled disk of radius $\frac{d}{2}$ (see ).

— Area and antipodal distance in convex hypersurfaces  (2604.02667 - Dibble et al., 3 Apr 2026) in Section 2 (When intrinsic distance is large relative to extrinsic distance)