Equivalence of Definitions of Convex Polyhedron

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:

For every face of $P$, the plane in which it is embedded does not intersect the interior of $P$.

Definition 3

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

For each face of $P$, the whole of $P$ lies on one side of the plane of that face.

Proof

$(1)$ implies $(2)$

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

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

$\Box$

$(2)$ implies $(1)$

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

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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.