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

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.

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.