Journal of Mathematical Physics, Analysis, Geometry
2020, vol. 16, No 3, pp. 291-311     ( to contents , go back )

Novel View on Classical Convexity Theory

Vitali Milman

Tel-Aviv University, Tel-Aviv, 69978, Israel

Liran Rotem

Technion - Israel Institute of Technology, Haifa, 32000, Israel

Received April 28, 2020.

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.


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

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