Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 3, pp. 291-311   https://doi.org/10.15407/mag16.03.291     ( to contents , go back )

### Novel View on Classical Convexity Theory

Vitali Milman

Tel-Aviv University, Tel-Aviv, 69978, Israel
E-mail: milman@tauex.tau.ac.il

Liran Rotem

Technion - Israel Institute of Technology, Haifa, 32000, Israel
E-mail: lrotem@technion.ac.il

Dedicated to the 100th birthday of the great convex geometer A.V. Pogorelov, to whom the first name author is infinitely grateful for very warm and useful meetings.

Abstract

Let $B_{x}\subseteq\mathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e., with with center at $\frac{x}{2}$ and radius $\frac{\left|x\right|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e., $F=\bigcup_{x\in A}B_{x}$ for any set $A\subseteq\mathbb{R}^{n}$. We showed earlier in [9] that the family of all flowers $\mathcal{F}$ is in 1-1 correspondence with $\mathcal{K}_{0}$ - the family of all convex bodies containing $0$. Actually, there are two essentially different such correspondences. We demonstrate a number of different non-linear constructions on $\mathcal{F}$ and $\mathcal{K}_{0}$. Towards this goal we further develop the theory of flowers.

Mathematics Subject Classification 2010: 52A20, 52A30, 52A23
Key words: convex bodies, owers, spherical inversion, duality, powers, Dvoretzky's Theorem