Width of spherical convex bodies

By M. Lassak. 2015

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For every hemisphere K supporting a convex body C on the sphere Sd we define the width of C determined by K. We show that it is a continuous function of the position of K. We prove that the diameter of every convex body $${C \subset S^d}$$C⊂Sd equals the maximum of the widths of C provided the diameter of C is at most $${\frac{\pi}{2}}$$π2. In a natural way, we define spherical bodies of constant width. We also consider the thickness Δ(C) of C, i.e., the minimum width of C. A convex body $${R \subset S^d}$$R⊂Sd is said to be reduced if Δ(Z) < Δ(R) for every convex body Z properly contained in R. For instance, bodies of constant width on Sd and regular spherical odd-gons of thickness at most $${\frac{\pi}{2}}$$π2 on S2 are reduced. We prove that every reduced smooth spherical convex body is of constant width.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9c5bb1acad04f9bc9851bbc9507e7038e020747d