Novel view on classical convexity theory

TL;DR

This paper introduces a new perspective on convexity by exploring the concept of flowers, unions of Euclidean balls called petals, and their correspondence with convex bodies, developing the theory further.

Contribution

It establishes new non-linear constructions on flowers and convex bodies, deepening the understanding of their relationship and expanding convexity theory.

Findings

Flowers are in 1-1 correspondence with convex bodies containing 0.

Develops new non-linear operations on flowers and convex bodies.

Enhances the theoretical framework connecting flowers and convexity.

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 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.