Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

Abstract

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 $\dim (C)$ of C, attaining 1 if and only if C is symmetric and $\dim (C)$ 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 $(1+\sqrt{5})/2\approx 1.618$, 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.

MSC:

• Primary 52A40
• Secondary 52A10

Acknowledgments

We would like to thank the anonymous referees for the useful suggestions that helped us to improve the article.

This research is partially a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia, and is partially funded by FEDER/Ministerio de Ciencia e Innovación - Agencia Estatal de Investigación. The third author is supported by Fundación Séneca project 19901/GERM/15, Spain, and by MICINN Project PGC2018-094215-B-I00 Spain.

Additional information

Funding

This research is partially a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia, and is partially funded by FEDER/Ministerio de Ciencia e Innovación - Agencia Estatal de Investigación. The third author is supported by Fundación Séneca project 19901/GERM/15, Spain, and by MICINN Project PGC2018-094215-B-I00 Spain.

Notes on contributors

René Brandenberg

RENÉ BRANDENBERG received his Ph.D. in mathematics from the Technical University of Munich after doing a diploma in applied mathematics at the University of Trier. After finishing the Ph.D., he joined the Technical University of Vienna before going back to the Technical University of Munich.

Katherina von Dichter

KATHERINA VON DICHTER graduated in mathematics at the Technical University of Munich and is now doing her Ph.D. supervised by the two coauthors.

Bernardo González Merino

BERNARDO GONZÁLEZ MERINO received his Ph.D. in mathematics from the University of Murcia. Afterwards, he joined the Technical University of Munich, the University Centre of Defence of San Javier, and the University of Seville as a postdoc until being appointed as an assistant professor at the University of Murcia in 2019.