Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

Similar to the arithmetic-harmonic mean inequality for numbers, the harmonic mean of two convex sets K and C is always contained in their arithmetic mean. The harmonic and arithmetic means of C and - C define two different symmetrizations of C, each keeping some useful properties of the original set...

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Bibliographic Details
Published in:The American mathematical monthly 2022-04, Vol.129 (4), p.352-362
Main Authors: Brandenberg, René, von Dichter, Katherina, Merino, Bernardo González
Format: Article
Language:English
Subjects:
Arithmetic
Asymmetry
Convexity
Harmonic analysis
Polynomials
Polytopes
Primary 52A40
Secondary 52A10
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Summary:Similar to the arithmetic-harmonic mean inequality for numbers, the harmonic mean of two convex sets K and C is always contained in their arithmetic mean. The harmonic and arithmetic means of C and - C define two different symmetrizations of C, each keeping some useful properties of the original set. We investigate the relations of such symmetrizations, involving a suitable measure of asymmetry-the Minkowski asymmetry, which, besides other advantages, is polynomial time computable for (reasonably given) polytopes. The Minkowski asymmetry measures the minimal dilatation factor needed to cover a set C by a translate of its negative. Its values range between 1 and the dimension of C, attaining 1 if and only if C is symmetric and if and only if C is a simplex. Restricting to planar compact, convex sets, positioned so that the translation in the definition of the Minkowski asymmetry is 0, we show that if the asymmetry of C is greater than the golden ratio , then the harmonic mean of C and - C is a subset of a dilatate of their arithmetic mean with a dilatation factor strictly less than 1; and for any asymmetry less than the golden ratio, there exists a set C with the given asymmetry value, such that the considered dilatation factor cannot be less than 1.
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2022.2043113