Shortest closed curve to contain a sphere in its convex hull

2022/09/13 by Ghomi, Mohammad, Wenk, James
#52A40 #58E #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 53A04 #Secondary: 60G15

paper · doi:10.48550/arxiv.2209.05988

Abstract

We show that in Euclidean 3-space any closed curve γ which contains the unit sphere within its convex hull has length L≥4π, and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in n dimensions, we include the estimate L≥ Cn√(n) by Nazarov, which is sharp up to the constant C.

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