User:MWinter4/Coordinate symmetric polytope

In geometry a subset $B\subseteq {\mathbb {R}}^{d}$ of Euclidean space is coordinate symmetric or unconditional if it is symmetric with respect to reflection on each of the coordinate hyperplanes $\{x\in {\mathbb {R}}^{d}\mid x_{i}=0\}$ for $i\in \{1,...,d\}$. Equivalently, $(x_{1},...,x_{d})\in B$ if and only if $(\epsilon _{1}x_{1},...,\epsilon _{d}x_{d})\in B$ for any choice of $\epsilon _{i}\in \{-1,+1\}$.

Coordinate symmetric sets are of particular interest in convex geometry and polytope theory, where one studies coordinate symmetric convex bodies and convex polytopes respectively. Those constitute a restricted yet surprisingly interesting class for which a number of claims have been verified that are open in general.

Properties

Coordinate symmetric convex bodies are always centrally symmetric and closed under a number of operations. If $H_{i}=\{x_{i}=0\}$ is the $i$-th coordinate hyperplane, then

• the section $B\cap H_{i}$ is coordinate symmetric.
• the projection $\pi _{i}(B)$ onto $H_{i}$ (i.e. essentially forgetting the $i$-th coordinate) is coordinate symmetric.
• projection and section yield the same result: $B\cap H_{i}=\pi _{i}(B)$.
• 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 commutes with both section and projection: $(B\cap H_{i})^{\circ }=B^{\circ }\cap H_{i}$ and $(\pi _{i}(B))^{\circ }=\pi _{i}(B^{\circ }).$

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),
• Godbersen's conjecture.

The coordinate symmetric polytope that are reflexive are in one to one relation with perfect graphs.

Antiblocking bodies

A coordinate symmetric corner or antiblocking polytope $P_{+}\subset {\mathbb {R}}_{+}^{d}$ 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.

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

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

Mahler's conjecture

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