Novel view on classical convexity theory

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 $\frac{|x|}{2}$. 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 in previous work 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.