A simple closed curve in \(\mathbb{R}^3\) whose convex hull equals the half-sum of the curve with itself (Q2188800)
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| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A simple closed curve in \(\mathbb{R}^3\) whose convex hull equals the half-sum of the curve with itself |
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A simple closed curve in \(\mathbb{R}^3\) whose convex hull equals the half-sum of the curve with itself (English)
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11 June 2020
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The author proves that, if \(\Gamma\) denotes the range of a rectifiable curve in \(\mathbb{R}^3\) and \(\mathrm{co}(\Gamma)\) is its convex hull, then \(\frac12(\Gamma + \Gamma)\) has zero Lebesgue measure in \(\mathbb{R}^3\); and he builds an example of a simple closed curve in \(\mathbb{R}^3\), whose range satisfies: \(\frac12(\Gamma + \Gamma) = \mathrm{co}(\Gamma) = [0, 1]^3\).
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simple closed curve
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Jordan curve
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Minkowski addition
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convex hull
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Peano curve
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