# Reciprocals and Flowers in Convexity

## TL;DR

This paper introduces new classes of convex and star bodies, such as reciprocal bodies and flowers, exploring their properties, relationships, and implications for convex geometry, including structure, operations, and inequalities.

## Contribution

It defines reciprocal bodies and flowers, studies their dualities and relations, and reveals their structural properties and implications for convex geometric operations and inequalities.

## Key findings

- Reciprocal bodies are characterized by their flowers being convex.
- The class of flowers is closed under Minkowski addition.
- Volume of flowers is a homogeneous polynomial in combination coefficients.

## Abstract

We study new classes of convex bodies and star bodies with unusual properties. First we define the class of reciprocal bodies, which may be viewed as convex bodies of the form "$1/K$". The map $K\mapsto K^\prime$ sending a body to its reciprocal is a duality on the class of reciprocal bodies, and we study its properties. To connect this new map with the classic polarity we use another construction, associating to each convex body $K$ a star body which we call its flower and denote by $K^\clubsuit$. The mapping $K\mapsto K^\clubsuit$ is a bijection between the class $\mathcal{K}_0^n$ of convex bodies and the class $\mathcal{F}^n$ of flowers. We show that the polarity map $\circ:\mathcal{K}_0^n\to\mathcal{K}_0^n$ decomposes into two separate bijections: First our flower map $\clubsuit:\mathcal{K}_0^n\to\mathcal{F}^n$, followed by the spherical inversion $\Phi$ which maps $\mathcal{F}^n$ back to $\mathcal{K}_0^n$. Each of these maps has its own properties, which combine to create the various properties of the polarity map. We study the various relations between the four maps $\prime$, $\circ$, $\clubsuit$ and $\Phi$ and use these relations to derive some of their properties. For example, we show that a convex body $K$ is a reciprocal body if and only if its flower $K^\clubsuit$ is convex. We show that the class $\mathcal{F}^n$ has a very rich structure, and is closed under many operations, including the Minkowski addition. This structure has corollaries for the other maps which we study. For example, we show that if $K$ and $T$ are reciprocal bodies so is their "harmonic sum" $(K^\circ+T^\circ)^\circ$. We also show that the volume $\left|\left(\sum_i\lambda_{i}K_i\right)^\clubsuit\right|$ is a homogeneous polynomial in the $\lambda_i$'s, whose coefficients can be called "$\clubsuit$-type mixed volumes". Related geometric inequalities are also derived.