A simple closed curve in ℝ³ whose convex hull equals the half-sum of the curve with itself

2018/07/22 by Patrakeev, Mikhail
#52A15 #53A04 #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1807.08257

Abstract

If Γ is the range of a Jordan curve that bounds a convex set in ℝ², then (1)/(2)(Γ+Γ)=co(Γ), where + is the Minkowski sum and co is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in ℝ³ with range Γ such that (1)/(2)(Γ+Γ)=[0,1]³=co(Γ). Also we show that such simple closed curve cannot be rectifiable.

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