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2020 A Simple Closed Curve in \(\mathbb{R}^3\) Whose Convex Hull Equals the Half-sum of the Curve with Itself
Mikhail Patrakeev
Real Anal. Exchange 45(1): 73-84 (2020). DOI: 10.14321/realanalexch.45.1.0073

Abstract

If \(\Gamma\) is the range of a Jordan curve that bounds a convex set in the plane, then \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma),\) where \(+\) is the Minkowski sum and \(\mathsf{co}\) is the convex hull. Answering a question of V. N. Ushakov, we construct a simple closed curve in \(\mathbb{R}^3\) whose range \(\Gamma\) satisfies \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma)=[0,1]^3.\) Also we show that such a simple closed curve cannot be rectifiable.

Citation

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Mikhail Patrakeev. "A Simple Closed Curve in \(\mathbb{R}^3\) Whose Convex Hull Equals the Half-sum of the Curve with Itself." Real Anal. Exchange 45 (1) 73 - 84, 2020. https://doi.org/10.14321/realanalexch.45.1.0073

Information

Published: 2020
First available in Project Euclid: 9 May 2020

zbMATH: 07211604
Digital Object Identifier: 10.14321/realanalexch.45.1.0073

Subjects:
Primary: 52A15 , 53A04
Secondary: 26A05

Keywords: Convex hull , Jordan curve , Minkowski addition , Peano curve , simple closed curve

Rights: Copyright © 2020 Michigan State University Press

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Vol.45 • No. 1 • 2020
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