User:MWinter4/Projectively unique polytope

In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations.

Introduction

Two convex polytope $P,Q\subset {\mathbb {R}}^{n}$ are said to be combinatorially equivalent if they have the same number of faces with the same incidence relations between them. Formally this can be expressed by an isomorphism between their face lattices. The following picture shows a number of polytopes that are combinatorially equivalent to the cube.

[PICTURE]

Given a convex polytope $P\subset {\mathbb {R}}^{n}$ many combinatorially equivalent polytopes can be generated through transformation of the ambient space, such as isometries or affine transformations. The largest class of transformations of Euclidean space that preserves coplanarity, and hence maps polytopes onto combiantorially equivalent polytopes, are projective transformations. A polytope $P$ is projectively unique if projective transformations are the only way to generate combinatorially equivalent polytopes; or in other words, if $P$ has a unique convex realization up to projective transformations.

Simplices, such as triangles and tetrahedra, are projectively unique. The cube is not projectively unique, as can seen from the following realization that is not projectively equivalent to the standard cube.

Projectively unique polytopes were first introduced by Peter McMullen, and later studied by Perles, Karim Adiprasito and Günter Ziegler.

Known projectively unique polytopes

A projective transformation on $n$-dimensional Euclidean space can map any $n+2$ points onto any other $n+2$ points. Hence, any $n$-dimensional polytope with at most $n+2$ vertices or facets is projectively unique. This includes simplices (which are even unique up to linear transformations). Suprisingly, there are projectively unique polytopes with more than $n+2$ vertices and facets. Moreover, being projectively unique is closed under polar duality.

Dimension two

The following is the complete list of projectively unique polytopes in dimension two:

• triangle
• quadrangle

Dimension three

A 3-polytope is projectively unique if and only if it has at most nine edges. The following is therefore the complete list of projectively unique polytopes in dimension three:

• tetrahedron
• square pyramid
• triangular prism
• triangular bipyramid

Dimension four

The following list of projectively unique 4-polytopes was compiled by Shepherd and is conjectured to be complete, though this is an open question:

#	construction	dual	f-vector	description
1	$\Delta _{4}$	1*	(5,10,10,5)	4-simplex
2	$\square \star \Delta _{1}$	2*	(6,11,11,6)	join of square and line segment
3	$(\Delta _{2}\oplus \Delta _{1})\star \Delta _{0}$	4	(6,14,15,7)	pyramid over a triangular bipyramid
4	$(\Delta _{2}\times \Delta _{1})\star \Delta _{0}$	3	(7,15,14,6)	pyramid over a triangular prism
5	$\Delta _{3}\oplus \Delta _{1}$	6	(6,14,16,8)	tetrahedral bipyramid
6	$\Delta _{3}\times \Delta _{1}$	5	(8,16,14,6)	tetrahedral prism
7	$\Delta _{2}\oplus \Delta _{2}$	8	(6,15,18,9)	cyclic polytope C(6,4)
8	$\Delta _{2}\times \Delta _{2}$	7	(9,18,15,6)	(3,3)-duoprism
9	$(\square ,v)\oplus (\square ,w)$	10	(7,17,18,8)	vertex sum of two squares
10	$((\square ,v)\oplus (\square ,w))^{\circ }$	9	(8,18,17,7)	the dual of $(\square ,v)\oplus (\square ,w)$
11	$(\Delta _{2}\times \Delta _{1},v)\oplus \Delta _{1}$	11*	(7,17,17,7)	subdirect sum of a triangular prism and a line segment at a vertex of the prism (this is also known as splitting the vertex)

* self-dual

Operations

McMullen showed that projectively unique polytopes are closed under any of the following operations:

• The join $P\star Q$ of two polytopes is projectively unique if and only if the $P$ and $Q$ are projectively unique.
• The vertex sum $(P,v)\oplus (Q,w)$ of two projectively unique polytopes is projectively unique.
• If $P$ is projectively unique and $v$ is a vertex of $P$ so that $P$ is not a vertex sum with distinguished vertex $v$, then $(P,v)\oplus \Delta _{1}$ is projectively unique.

These opertaions are sufficient to generate Shephards list of projectively unique 4-polytopes.

Other examples

While there are only finitely many projectively unique polytopes in dimension two and three (and this is conjectured to be the case also in dimension four), Adiprasito & Ziegler (2015) constructed infinitely many combinatorially distinct projectively unique polytopes of dimension 69.

Linearly unique polytopes

A centrally symmetric polytope $P$ is linearly unique (or linearly stable) if every realization of $P$ as a centrally symmetric polytope is linearly equivalent to $P$. Like for projective uniqueness, the class of linear unique polytopes is closed under polar duality (where one assumes that the polarity center is chosen at the symmetry center of the polytope, so that the polar dual is centrally symmetric as well).

In 1969 Peter McMullen showed that every centrally symmetric 0/1-polytope is linearly unique. In the same paper McMullen claims that this gives a complete characterization of linearly unique polytopes. This was later found to be incorrect by David Assaf (1976). The concrete counterexample he provided is the polar dual of a centrally symmetic 0/1-polytopes that is itself not a 0/1-polytopes.