Optimal constant and extremal bodies in the Pál–Firey minimum‑width volume inequality for d ≥ 3

Determine the optimal constant F_d and characterize the extremal convex sets in R^d, d ≥ 3, for the Pál–Firey inequality asserting that any compact convex set Ω with minimum width w satisfies Vol(Ω) ≥ F_d w^d.

Background

The Pál–Firey theorem gives a universal lower bound on volume in terms of minimum width; the sharp constant and extremizers are known in the planar case (d=2), where equality occurs for equilateral triangles.

In higher dimensions, it is known that the Euclidean ball is not optimal, and recent work provides improved bounds, but the exact optimal constant and the equality cases remain unresolved. Progress on this question influences volume–width inequalities used in the paper’s lower bounds for convex hypersurfaces.

References

The sharp version of Theorem \ref{pal-firey} is known only for $d = 2$, but, although the optimal constant and domain are not known for $d \geq 3$, it is known that the ball is not optimal: a cone of height $1$ over a $(d-1)$-disk of radius $\frac{1}{\sqrt{3}$ contains a unit-length line segment in every direction and has smaller volume than the $d$-ball of radius $\frac{1}{2}$.

— Area and antipodal distance in convex hypersurfaces  (2604.02667 - Dibble et al., 3 Apr 2026) in Remark on Pál–Firey (Section 3)

The classical conjecture, generally attributed to Bonnesen and Fenchel, asserts that the minimum is attained by the Meissner tetrahedra; see . This remains one of the central open problems concerning bodies of constant width.

— Dangling points in area-minimizing Meissner polyhedra  (2608.19747 - Bogosel, 20 Aug 2026) in Section 1, Introduction