User:MWinter4/Coordinate symmetric polytope

In geometry, a convex body is unconditional (also 1-unconditional or coordinate symmetric) if it is symmetric with respect to reflection on each coordinate hyperplane. Equivalently, a body ${\displaystyle K\subseteq {\mathbb {R}}^{n}}$ is unconditional if from ${\displaystyle (x_{1},...,x_{n})\in K}$ follows ${\displaystyle (\pm x_{1},...,\pm x_{n})\in K}$ for each choice of the signs. Unconditional bodies constitute a restricted yet surprisingly interesting class for which a number of claims have been verified that are open in general.

The term "unconditional" originates from unconditional normed spaces whose unit balls are precisely the unconditional convex bodies. A normed spaces ${\displaystyle E}$ is unconditional if there exists an unconditional basis ${\displaystyle e_{1},...,e_{n}\in E}$, that is,

${\displaystyle \displaystyle {\Big \|}\sum _{i}a_{i}e_{i}{\Big \|}={\Big \|}\sum _{i}|a_{i}|e_{i}{\Big \|}.}$

or in other words, the series converges unconditionally.^[1]

Properties

Unconditional bodies are centrally symmetric and are closed under a number of operations. Let ${\displaystyle H_{i}=\{x\in {\mathbb {R}}^{n}\mid x_{i}=0\}}$  be the ${\displaystyle i}$ -th coordinate hyperplane. Then

• the section ${\displaystyle B\cap H_{i}}$  is coordinate symmetric.
• the orthogonal projection ${\displaystyle \pi _{i}(B)}$  onto ${\displaystyle H_{i}}$  is coordinate symmetric (and is identical to the section, that is, ${\displaystyle B\cap H_{i}=\pi _{i}(B)}$ ).
• the polar dual ${\displaystyle B^{\circ }:=\{x\in {\mathbb {R}}^{d}\mid \langle x,y\rangle \leq 1}$  for all ${\displaystyle y\in B\}}$  is coordinate symmetric.

Polar duality also commutes with both section and projection: ${\displaystyle (B\cap H_{i})^{\circ }=B^{\circ }\cap H_{i}}$  and ${\displaystyle (\pi _{i}(B))^{\circ }=\pi _{i}(B^{\circ })}$ . Here the polar dual of a body contained in ${\displaystyle H_{i}}$  is understood to be restricted to ${\displaystyle H_{i}}$  as well.

Unconditional bodies are a common test case for conjectures in convex geometry. Some famously open conjecture have been verified for unconditional bodies. This includes

• Mahler conjecture,
• Kalai's 3^d conjecture (for unconditional polytopes), as well as the related full flag conjecture,
• Godbersen's conjecture.

The coordinate symmetric reflexive polytopes are in one-to-one relation with perfect graphs.

Positive corners

An unconditional body ${\displaystyle K\subseteq {\mathbb {R}}^{n}}$  is determined by its restriction ${\displaystyle K_{+}:=K\cap {\mathbb {R}}_{+}^{n}}$  to the positive orthant ${\displaystyle {\mathbb {R}}_{+}^{n}:=\{x\in {\mathbb {R}}^{n}\mid x_{i}\geq 0\}}$ . The restriction ${\displaystyle K_{+}}$  of an uncondition body to the positive orthant is called a positive corner (or anti-blocking body). Unconditional bodies are in one-to-one correspondence with positive corners.

An alternative but self-contained definition of anti-blocking body is the following: a subset ${\displaystyle B_{+}\subseteq {\mathbb {R}}_{+}^{n}}$  is an anti-blocking body if for each ${\displaystyle x\in B_{+}}$  and ${\displaystyle y\in {\mathbb {R}}_{+}^{n}}$  with ${\displaystyle y_{i}\leq x_{i}}$  (for each coordinate) holds ${\displaystyle y\in B}$ .

The term "anti-blocking" is derived from the notion of (anti-)blocking in hereditary set systems.^[2]

An anti-blocking body (or convex corner) ${\displaystyle B_{+}\subset {\mathbb {R}}_{+}^{d}}$  is the restriction of a coordinate symmetric polytope to the positive orthant, that is, ${\displaystyle P_{+}=P\cap {\mathbb {R}}_{+}^{d}}$ . Like unconditional polytopes, antiblocking polytopes are closed under projection onto and section with coordinate hyperplanes. Moreover, projection and section yield the same polytopes.

${\displaystyle B_{+}^{*}:=\{x\in {\mathbb {R}}_{+}^{n}\mid \langle x,y\rangle \leq 1{\text{ for all }}y\in B_{+}\}.}$ 

If ${\displaystyle B_{+}=B\cap {\mathbb {R}}_{+}^{d}}$ , then the antiblocking dual of ${\displaystyle B_{+}}$  is ${\displaystyle B_{+}^{*}:=B^{\circ }\cap {\mathbb {R}}_{+}^{d}}$ . As for unconditional polytopes, this notion of duality commutes with section and projection.

Mahler conjecture

The statement of the Mahler conjecture for antiblocking polytopes reduces to

${\displaystyle M_{+}(P_{+}):=\operatorname {vol} (P_{+})\cdot \operatorname {vol} (P_{+}^{*})\geq 1/d!}$ .

A fairly short inductive proof goes as follows: let ${\displaystyle P_{i}:=P_{+}\cap \{x_{i}=0\}}$  and ${\displaystyle P_{i}^{*}:=P_{+}^{*}\cap \{x_{i}=0\}}$ , which are antiblocking. Define vectors ${\displaystyle v,v^{*}\in {\mathbb {R}}^{d}}$  by

${\displaystyle v_{i}:={\frac {\operatorname {vol} _{d-1}(P_{i})}{d\operatorname {vol} _{d}(P_{+})}},\quad \;v_{i}^{*}:={\frac {\operatorname {vol} _{d-1}(P_{i}^{*})}{d\operatorname {vol} _{d}(P_{+}^{*})}}.}$ 

For a point ${\displaystyle x\in {\mathbb {R}}_{+}^{d}}$  the inner product ${\displaystyle \langle x,v\rangle }$  evaluates to

${\displaystyle \langle x,v\rangle =\sum _{i=1}^{d}x_{i}{\frac {\operatorname {vol} _{d-1}(P_{i})}{d\operatorname {vol} _{d}(P_{+})}}={\frac {1}{\operatorname {vol} _{d}(P_{+})}}\cdot \sum _{i=1}^{d}\underbrace {{\tfrac {1}{d}}{x_{i}\operatorname {vol} _{d-1}(P_{i})}} _{\operatorname {vol} _{d}(C_{i})},}$ 

where ${\displaystyle C_{i}}$  is the cone with base face ${\displaystyle P_{i}}$  and cone point ${\displaystyle x}$ . If ${\displaystyle x\in P_{+}}$  then the cones ${\displaystyle C_{i}}$  have disjoint interiors and are contained in ${\displaystyle P_{+}}$ . In particular, the sum of their volumes is bounded by ${\displaystyle \operatorname {vol} _{d}(P_{+})}$ . Therefore ${\displaystyle \langle x,v\rangle \leq 1}$  for all ${\displaystyle x\in P_{+}}$ , and hence ${\displaystyle v\in P_{+}^{*}}$ . By an analogous argument holds ${\displaystyle v^{*}\in P_{+}}$ . In particular,

${\displaystyle {\begin{aligned}1\geq &\langle v,v^{*}\rangle =\sum _{i=1}^{d}{\frac {\operatorname {vol} _{d-1}(P_{i})\operatorname {vol} _{d-1}(P_{i}^{*})}{d^{2}\operatorname {vol} _{d}(P_{+})\operatorname {vol} _{d}(P_{+}^{*})}}=\sum _{i=1}^{d}{\frac {M_{+}(P_{i})}{d^{2}M_{+}(P_{+})}}.\end{aligned}}}$ 

By rearranging and applying the induction hypothesis ${\displaystyle M_{+}(P_{i})\geq 1/(d-1)!}$  we conclude

${\displaystyle M_{+}(P_{+})\geq {\frac {1}{d^{2}}}\sum _{i=1}^{d}M_{+}(P_{i})\geq {\frac {1}{d^{2}}}\cdot {\frac {d}{(d-1)!}}={\frac {1}{d!}}.}$ 

Locally anti-blocking bodies

A convex body is locally anti-blocking or locally coordinate symmetric if its restriction to any orthant (after rotation to the positive orthant) is antiblocking. Locally anti-blocking bodies are not necessarily centrally symmetric, and conversely, not every centrally symmetric body is locally antiblocking.

Most properties of unconditional bodies translate to locally antiblocking bodies: locally antiblocking bodies are closed under projection onto and section with coordinate hyperplanes, and projection and section yield the same result. Locally antiblocking bodies are closed under polarity, and polarity commutes with projection and section. Also many results for coordinate symmetric polytopes generalize: the Mahler conjecture is proven for locally antblocking bodies, and Kalai's 3^d conjecture is proven for locally antiblocking polytopes.

Locally antiblocking bodies can be equally characterized as follows: if ${\displaystyle x\in B}$  and ${\displaystyle n\in N_{B}(x)}$  (where ${\displaystyle N_{B}(x)}$  is the normal cone of ${\displaystyle B}$  at ${\displaystyle x}$ ) then ${\displaystyle x_{i}n_{i}\geq 0}$  in each component.

In the case of polytopes, it is not well understood which centrally symmetric polytopes have a locally antiblocking realization.

Mahler's conjecture

Define ${\displaystyle x_{i}:=\operatorname {vol} (P_{i}^{+})/d\operatorname {vol} (P^{+})}$  ...

Kalai's 3^d conjecture

References

1. Heil, C. (2010). A basis theory primer: expanded edition. Springer Science & Business Media. Chapter 6
2. Section 2.1