Schlegel diagram

Definition

Let $P \subset \R^d$ be a convex polytope of dimension $d \ge 2$ , and let $F$ be a facet of $P$ . For each facet $G$ , let $H_G$ be the hyperplane spanned by $G$ . Choose a point $p \in \R^d$ such that:

$p$ and $P$ lie on opposite sides of $H_F$ , strictly;
for every facet $G \ne F$ , $p$ lies strictly on the same side of $H_G$ as the interior of $P$ .

For $x \in P$ , let $\pi(x)$ be the point where the segment from $p$ to $x$ meets $H_F$ . The Schlegel diagram of $P$ based at $F$ is the collection of the sets $\pi(G')$ . Here $G'$ runs over the faces of $P$ other than $P$ and $F$ .

Remarks

Each $\pi(G')$ is a convex polytope contained in $F$ , of the same dimension as $G'$ . These polytopes cover $F$ , two of them meet in a common face, and their inclusions are those of the faces of $P$ . So the diagram is a polytopal subdivision of $F$ that records the whole face lattice of $P$ . Points $p$ as in the definition exist; any point beyond $F$ close enough to an interior point of $F$ works.

Example: let $P = [0,1]^3$ , $F$ the facet $x_3 = 1$ , and $p = (\tfrac12, \tfrac12, 1 + \epsilon)$ with $\epsilon > 0$ small. Then $p$ is above $x_3 = 1$ , and on the inner side of the four side planes and of $x_3 = 0$ . The bottom square projects to a smaller square in the middle of $F$ , and the four side squares project to trapezoids between the two squares.

Non-example: in the same example, $p = (5, \tfrac12, 2)$ satisfies condition 1 but fails condition 2 for the facet $x_1 = 1$ , since $p$ has $x_1 > 1$ . Then the images are not even inside $F$ : the segment from $p$ to the vertex $(1, 0, 0)$ meets the plane $x_3 = 1$ at $(3, \tfrac14, 1) \notin F$ .

Tag	`BFV3MKCBHZ`
URI	`m2://BFV3MKCBHZ`
Kind	definition
Writing	W0 stub
MSC	52B05, 52B11, 52B10
Defines	Schlegel diagram
Uses	Convex polytope, Facet, Face of a convex set, Affine span

Schlegel diagram

Definition

Remarks

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