User:MWinter4/Adjoint of a polytope

In geometry, and specifically in polytope theory, the adjoint of a convex polytope P is a particular polynomial associated to P. There are several equivalent definitions whose equivalence is not immediately obvious.

Dual volumes

Let ${\displaystyle P\subset {\mathbb {R}}^{d}}$  be a convex polytope. For a point ${\displaystyle x\in {\mathbb {R}}^{d}}$  one can associate the polar dual polytope

${\displaystyle (P-x)^{\circ }=\{y\in {\mathbb {R}}^{d}\mid \langle z-x,y\rangle \leq 1{\text{ for all }}z\in P\}.}$ 

It turns out that the volume of ${\displaystyle (P-x)^{\circ }}$  extends to a rational function ${\displaystyle \Omega _{P}\in {\mathbb {R}}(x)}$  on all of ${\displaystyle {\mathbb {R}}^{d}}$  with poles precisely along the facet hyperplanes of ${\displaystyle P}$ . The expression

${\displaystyle \operatorname {adj} _{P}(x):=\operatorname {vol} (P-x)^{\circ }\prod _{F}\ell _{F}(x)}$ 

is therefore a polynomial. Here the product is taken over all facets ${\displaystyle F}$  of ${\displaystyle P}$  where ${\displaystyle \ell _{F}(x):=b_{F}-\langle x,u_{F}\rangle }$  is the defining linear form with ${\displaystyle u_{F}}$  an outwards pointing normal vector.

Triangulations and explicit expression

${\displaystyle \operatorname {adj} _{P}(x)=\sum _{\sigma \in {\mathcal {T}}}\operatorname {vol} (\sigma )\prod _{F\not \sim \sigma }\ell _{F}(x).}$ 

Residual arrangement and adjoint hypersurface

The hyplerplane arrangement of a polytope ${\displaystyle P}$  consists of all the hyperplanes defined by faces of ${\displaystyle P}$  together with the affine subspaces defined through their intersection. The residual arrangement is the sub-arrangement obtained by removing the subspaces spanned by faces of ${\displaystyle P}$ . It turns out that the adjoint vanishes on all points of the residual arrangement. Due to Kohn & Ranestad the adjoint is the lowest degree polynomial with this property. The vanishing set of the adjoint is knowsn as the adjoint hypersurface.

Canonical forms and valuations

The rational function

${\displaystyle \Omega _{P}(x)={\frac {\operatorname {adj} _{P}(x)}{\prod _{F}\ell _{F}(x)}}=\operatorname {vol} (P-x)^{\circ }}$ 

is known as the canonical function of the polytope. The remarkable fact is that it behaves as a rational-function valued valuation on convex polytopes.

Integral expression

${\displaystyle \operatorname {adj} _{P}(x)=\int _{{\mathbb {R}}^{d}}\exp(h_{P}(x)-\langle x,y\rangle )\,\mathrm {d} y}$ 

Wachspress coordinates

The adjoint can be used to define the Wachspress coordinates:

${\displaystyle w_{v}(x):={\frac {...}{\operatorname {adj} _{P}(x)}}.}$