# The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Problem Statement and Our Contribution

## 2. Preliminaries

#### 2.1. Notations and Definitions

#### 2.2. Problem Formulations

## 3. The Existence of a Convex Polygon in the Plane

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Case 1**$\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=4$

**Case 2**$\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=2$ or 3

**Theorem**

**2.**

**Proof.**

**Case 1**$\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=4$

**Case 2**$\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=2$ or 3

## 4. Existence of a Convex Polyhedron in Three-Dimensional Space

**Lemma**

**3.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Applications

**Theorem**

**4.**

**Proof.**

## 6. Concluding Remarks

**Conjecture:**For any set of given constrained vertex norms$\mathcal{R}$, it is not always possible to find the convex configuration with respect to the given set$\mathcal{R}$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) The general configuration of points on the concentric circles that forms a non-convex polygon; (

**right**) the convex configuration of points with respect to the constrained vertex norms.

**Figure 2.**The construction of a convex polygon with respect to the given distinct constrained vertex norms.

**Figure 3.**The construction for the convex configuration of V when (

**left**) $\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=4$ and (

**right**) $\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=2$ or 3.

**Figure 4.**The perturbation of the points to find the strictly convex configuration when $\mathrm{max}\{{m}_{1},\dots ,{m}_{k}\}=4$.

**Figure 5.**The cross section at the $YZ$-plane for the concentric spheres including a spherical circle of each layer and the cone $\mathcal{C}$.

**Figure 6.**The Voronoi diagrams of 50 generators; (

**left**) the ordinary spherical Voronoi diagram; (

**right**) the spherical Laguerre Voronoi diagram with random weights. Certain cells lose in the case of the spherical Laguerre Voronoi diagram.

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**MDPI and ACS Style**

Chaidee, S.; Sugihara, K. The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms. *Mathematics* **2020**, *8*, 645.
https://doi.org/10.3390/math8040645

**AMA Style**

Chaidee S, Sugihara K. The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms. *Mathematics*. 2020; 8(4):645.
https://doi.org/10.3390/math8040645

**Chicago/Turabian Style**

Chaidee, Supanut, and Kokichi Sugihara. 2020. "The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms" *Mathematics* 8, no. 4: 645.
https://doi.org/10.3390/math8040645