Abstract
In this note we discuss new constructions of convex bodies. By thinking of the polarity map K ↦ K ∘ as the inversion x ↦ x −1 one may construct new bodies which were not previously considered in convex geometry. We illustrate this philosophy by describing a recent result of Molchanov, who constructed continued fractions of convex bodies.
Our main construction is the geometric mean of two convex bodies. We define it using the above ideology, and discuss its properties and its structure. We also compare our new definition with the “logarithmic mean” of Böröczky, Lutwak, Yang and Zhang, and discuss volume inequalities. Finally, we discuss possible extensions of the theory to p-additions and to the functional case, and present a list of open problems.
An appendix to this paper, written by Alexander Magazinov, presents a 2-dimensional counterexample to a natural conjecture involving the geometric mean.
The research was conducted while the author worked in Tel-Aviv University.
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Acknowledgements
The research was supported by ISF grant 826/13 and BSF grant 2012111. The second named author was also supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. We would like to thank the referee for his/her useful suggestions.
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Milman, V., Rotem, L. (2017). Non-standard Constructions in Convex Geometry: Geometric Means of Convex Bodies. In: Carlen, E., Madiman, M., Werner, E. (eds) Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7005-6_12
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DOI: https://doi.org/10.1007/978-1-4939-7005-6_12
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