#
A New Approach to Circular Inversion in l_{1}-Normed Spaces

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**i.**$\rho \left(\alpha \right)\ge 0,\phantom{\rule{4pt}{0ex}}\forall \alpha \in V\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$(positive definite, $(\rho \left(\alpha \right)=0\phantom{\rule{4pt}{0ex}}iff\phantom{\rule{4pt}{0ex}}\alpha =\overrightarrow{0})$);

**ii.**$\rho \left(\lambda \alpha \right)=\left|\lambda \right|\rho \left(\alpha \right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$(positive homogeneity);

**iii.**$\rho \left(\alpha +\beta \right)\le \rho \left(\alpha \right)+\rho \left(\beta \right)$ (triangle inequality).

**Lemma**

**1.**

**Proof.**

**(i)**The unit ball is a closed and bounded set;

**(ii)**The unit ball is centrally symmetric;

**(iii)**The unit ball is convex.

**i**,

**ii**, and

**iii**, then the function

## 2. Literature Review and Motivation

**i.**The inverse of a circle (not through the center of inversion) is a circle;

**ii.**The inverse of a circle through the center of inversion is a line;

**iii.**The inverse of a line (not through the center of inversion) is a circle through the center of inversion;

**vi.**A circle orthogonal to the circle of inversion is its own inverse;

**v.**A line through the center of inversion is its own inverse;

**vi.**angles are preserved under inversion.

**v**, remain invariant in the normed spaces $\left({\mathbb{R}}^{n},{\u2225.\u2225}_{p}\right)$ and $\left({\mathbb{R}}^{n},{\u2225.\u2225}_{\alpha}\right)$ (see [31,33] for more details).

## 3. A Visit to Circular Inversion in Minkowski Geometries $({\mathbb{R}}^{2},{l}_{1})$ and $({\mathbb{R}}^{2},{l}_{2})$

**Definition**

**1.**

**Simple Construction:**To construct the inverse ${P}^{|}$ of a point P outside the circle $C$ (see Figure 4):

## 4. Main Result

#### 4.1. Notation and Preliminary

**Lemma**

**2.**

#### 4.2. Synthetic Construction of Circular Inversion in ${l}_{1}$-Normed Space

**Theorem**

**1**

**.**Let ${C}_{T}$ be a ${l}_{1}$-normed circle with the radius ${r}_{T}$ and center O in the Minkowski plane $({\mathbb{R}}^{2},{l}_{1})$. Then, the image of a point P under inversion with respect to ${C}_{T}$ is

## 5. Conclusions

**Open problem:**In Minkowski geometry with an ${l}_{2}$-norm (Euclidean geometry), the image of a point under circular inversion can be found by a synthetic method. Similarly, can a synthetic method be constructed to find the image of a point under circular inversion in Minkowski geometry (geometry that emerges through norms in a finite-dimensional Banach space)?

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Huernia flowers and how the outer edges of the corolla (blue) are obtained by the inversion of the sepals (light blue) over the green circle k (see [11] for details).

**Figure 3.**Unit circles in terms of the norms ${\u2225\left(x,y\right)\u2225}_{2,\infty}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\u2225\left(x,y\right)\u2225}_{\infty ,1}$, respectively.

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**MDPI and ACS Style**

Ermiş, T.; Şen, A.O.; Gielis, J.
A New Approach to Circular Inversion in *l*_{1}-Normed Spaces. *Symmetry* **2024**, *16*, 874.
https://doi.org/10.3390/sym16070874

**AMA Style**

Ermiş T, Şen AO, Gielis J.
A New Approach to Circular Inversion in *l*_{1}-Normed Spaces. *Symmetry*. 2024; 16(7):874.
https://doi.org/10.3390/sym16070874

**Chicago/Turabian Style**

Ermiş, Temel, Ali Osman Şen, and Johan Gielis.
2024. "A New Approach to Circular Inversion in *l*_{1}-Normed Spaces" *Symmetry* 16, no. 7: 874.
https://doi.org/10.3390/sym16070874