# Equivalence of Definitions of Convex Polyhedron

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## Theorem

The following definitions of the concept of **Convex Polyhedron** are equivalent:

### Definition 1

$P$ is a **convex polyhedron** if and only if:

- For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$.

### Definition 2

$P$ is a **convex polyhedron** if and only if:

### Definition 3

$P$ is a **convex polyhedron** if and only if:

## Proof

### $(1)$ implies $(2)$

Let $P$ be a convex polyhedron by definition $1$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Thus $P$ is a convex polyhedron by definition $2$.

$\Box$

### $(2)$ implies $(1)$

Let $P$ be a convex polyhedron by definition $2$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Thus $P$ is a convex polyhedron by definition $1$.

$\Box$

This needs considerable tedious hard slog to complete it.In particular: definition 3 etc.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |