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 $K\subseteq {\mathbb {R}}^{n}$ is unconditional if from $(x_{1},...,x_{n})\in K$ follows $(\pm x_{1},...,\pm x_{n})\in K$ 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 $E$ is unconditional if there exists an unconditional basis $e_{1},...,e_{n}\in E$ , that is,

\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 $H_{i}=\{x\in {\mathbb {R}}^{n}\mid x_{i}=0\}$ be the $i$ -th coordinate hyperplane. Then

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

Polar duality also commutes with both section and projection: $(B\cap H_{i})^{\circ }=B^{\circ }\cap H_{i}$ and $(\pi _{i}(B))^{\circ }=\pi _{i}(B^{\circ })$ . Here the polar dual of a body contained in $H_{i}$ is understood to be restricted to $H_{i}$ 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 $K\subseteq {\mathbb {R}}^{n}$ is determined by its restriction $K_{+}:=K\cap {\mathbb {R}}_{+}^{n}$ to the positive orthant ${\mathbb {R}}_{+}^{n}:=\{x\in {\mathbb {R}}^{n}\mid x_{i}\geq 0\}$ . The restriction $K_{+}$ 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 $B_{+}\subseteq {\mathbb {R}}_{+}^{n}$ is an anti-blocking body if for each $x\in B_{+}$ and $y\in {\mathbb {R}}_{+}^{n}$ with $y_{i}\leq x_{i}$ (for each coordinate) holds $y\in B$ .

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

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

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

If $B_{+}=B\cap {\mathbb {R}}_{+}^{d}$ , then the antiblocking dual of $B_{+}$ is $B_{+}^{*}:=B^{\circ }\cap {\mathbb {R}}_{+}^{d}$ . 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

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

.

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

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

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

{\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 $M_{+}(P_{i})\geq 1/(d-1)!$ we conclude

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 $x\in B$ and $n\in N_{B}(x)$ (where $N_{B}(x)$ is the normal cone of $B$ at $x$ ) then $x_{i}n_{i}\geq 0$ in each component.