Approximation of Spherical Convex Bodies of Constant Width π/2

Let

be the unit sphere in

\mathbb{R}^3 R 3

and let

C\subset \mathbb{S}^2 C ⊂ S 2

be a spherical convex body of constant width

\tau τ

. It is known that (i) if

\tau<\pi/2 τ < π / 2

then for any

\varepsilon>0 ε > 0

there exists a spherical convex body

C_\varepsilon C ε

of constant width

\tau τ

whose boundary consists only of arcs of circles of radius

\tau τ

such that the Hausdorff distance between

C C

and

C_\varepsilon C ε

is at most

\varepsilon ε

; (ii) if

\tau>\pi/2 τ > π / 2

then for any

\varepsilon>0 ε > 0

there exists a spherical convex body

C_\varepsilon C ε

of constant width

\tau τ

whose boundary consists only of arcs of circles of radius

\tau-\frac{\pi}{2} τ − π 2

and great circle arcs such that the Hausdorff distance between

C C

and

C_\varepsilon C ε

is at most

\varepsilon ε

. In this paper, we present an approximation of the remaining case

\tau=\pi/2 τ = π / 2

, that is, if

\tau=\pi/2 τ = π / 2

then for any

\varepsilon>0 ε > 0

there exists a spherical polygon

\mathcal{P}_\varepsilon P ε

of constant width

\pi/2 π / 2

such that the Hausdorff distance between

C C

and

\mathcal{P}_\varepsilon P ε

is at most

\varepsilon ε

.