Approximation by polytopes of spherical constant-width π/2 bodies in S^n

Prove that for any dimension n ≥ 2 and any spherical convex body C ⊂ S^n of constant width π/2, for every ε > 0 there exists a spherical convex polytope P_ε ⊂ S^n of constant width π/2 such that the Hausdorff distance between C and P_ε is at most ε.

Background

The paper studies approximation of spherical convex bodies of constant width on the unit sphere. Earlier results established that if the width τ is less than π/2, then any such body can be approximated by bodies whose boundaries are arcs of circles of radius τ, and if τ exceeds π/2, by bodies whose boundaries are arcs of circles of radius τ−π/2 and great circle arcs. The present paper resolves the remaining two-dimensional case τ = π/2 by proving that any spherical convex body of constant width π/2 in S^2 can be approximated by spherical polytopes of the same constant width.

The authors explicitly reference a conjecture formulated in prior work that asserts the same approximation property in S^n for general dimension. Their Theorem 3 confirms the conjecture for n = 2, leaving the higher-dimensional cases open.