When is a spherical body of constant diameter of constant width?

Abstract

Let D be a convex body of diameter δ, where 0 < δ < π 2 , on the d-dimensional sphere. We prove that D is of constant diameter δ if and only if it is of constant width δ in the following two cases. The first case is when D is smooth. The second case is when d = 2.

Complete spherical convex bodies

“…By [4] the inverse holds for any δ. Our short proof of Theorem 3 is quite different from the considerations in [8], [4] and [7]. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.…”

Diameter, width and thickness in the hyperbolic plane

“…• Both the track function and the intersection function are continuous on each interval (0; π) and (π; 2π) and for θ ∈ (0, π) hold that x(θ+π) = x(θ), 0 ≤ f (θ) ≤ d and f (θ) + f (θ + π) = d. • The diametral chord AC can be chosen in such away that the continuity property of these functions extended to the interval (0, 2π) giving that There is a nice recent result of M. Lassak in [10] which connects the spherical concepts of constant width and constant diameter. He proved that a convex body on the two-dimensional sphere of diameter δ ≤ π/2 is of constant diameter δ if and only if it is of constant width δ.…”

Diameter, width and thickness in the hyperbolic plane

“…• The diametral chord AC can be chosen in such away that the continuity property of these functions extended to the interval (0, 2π) giving that they are can be extended periodically such that the above properties remain valid on R. There is a nice recent result of M. Lassak in [11] which connects the spherical concepts of constant width and constant diameter. He proved that a convex body on the two-dimensional sphere of diameter δ is less than π/2 is of constant diameter δ if and only if it is of constant width δ.…”