Secondary Power Diagram, Dual of Secondary Polytope

Abstract

An ingenious construction of Gel’fand et al. (Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel, 1994) geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with non-empty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a non-degenerated coherent triangulation to another non-degenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are non-degenerated and coherent.

Appendix

Notation
y i	A point in $\mathbb {R}{ }^n$
Y	Point configuration, Y = {y 1, y 2, …, y k}
T	A triangulation of (Y, Conv(Y ))
Σ(Y )	Secondary polytope, all coherent triangulations
$\mathcal {T}(Y)$	All non-degenerated coherent triangulations
π i	Hyperplane π i(x) = 〈x, y i〉− h i
π i ∗	Dual point of π i, $\pi _i^*=(y_i,h_i)$
Env(h)	Upper envelope of π i’s, Env({π i})
Conv(h)	Convex hull of $\pi _i^*$, $Conv(\{\pi _i^*\})$
u h(x)	Upper envelope function u h(x) =maxi π i(x)
$u_h^*(y)$	Legendre dual of u h(x)
D(h)	Power diagram by projecting Env(h)
Ω	A convex domain in $\mathbb {R}{ }^n$
w i(h)	The volume of the intersection between Ω and a cell of D(h)
T(h)	Weighted Delaunay triangulation by projecting Conv(h)
$\mathcal {H}_\varOmega ^\varepsilon (Y)$	Generalized Alexandrov power diagram space with respect to (Ω, ε),
	ε ≥ 0, the space of all power diagrams D(h)’s parameterized by h,
$\overline {\mathcal {H}}_\varOmega ^\varepsilon (Y)$	The closure of $\mathcal {H}_\varOmega ^\varepsilon (Y)$
	all w i(h)’s are positive, boundary cell volumes are greater than ε
Π(Y )	Secondary power diagram, the power diagram of $\overline {\mathcal {H}}_\varOmega ^\varepsilon (Y)$,
	which is dual to Σ(Y ).

1.1 I Proof of Theorem 21 Convexity of Secondary Polytope

Corollary 24 (Convexity of Secondary Polytope)

Suppose each coherent triangulation T of (Y, Conv(Y )) is represented as its characteristic function ψ _T , then all the ψ _T ’s are on their convex hull Conv({ψ _T}).

Proof

The upper envelope of the power diagram of $\overline {\mathcal {H}}_\varOmega ^\varepsilon (Y)$ is Env({π _T}), where each supporting plane π _T(h) = 〈ψ _T, h〉, therefore its dual point $\pi _T^*$ is (ψ _T, 0). The upper envelope Env({π _T}) is the graph of the piece-wise linear function U(Y ). The Legendre dual of U(Y ) is U ^∗(Y ), whose graph is the convex hull $Conv(\{\pi _T^*\})$,

$$\displaystyle \begin{aligned} Conv(\{\pi_T^*\}) = Conv(\{(\psi_T,0)\})=Conv(\{\psi_T\}). \end{aligned}$$

Due to the convexity of U(Y ) and Legendre dual of a convex function is convex, therefore ψ _T is on the convex hull $Conv(\{\pi _T^*\})$.

1.2 I Proof of Theorem 22 Complete Fan Structure

Corollary 25 (Complete Fan Structure)

• 1. Suppose 1 < λ, then

  $$\displaystyle \begin{aligned} \lambda\overline{\mathcal{H}}_{\varOmega}^\varepsilon = \overline{\mathcal{H}}_{\lambda\varOmega}^{\lambda^n\varepsilon} \subset \overline{\mathcal{H}}_{\lambda\varOmega}^\varepsilon , \end{aligned}$$

  for each coherent triangulation T ∈ Σ(Y ), the cell $D_T(\varOmega ,\varepsilon ) \subset \overline {\mathcal {H}}_{\varOmega }^\varepsilon$ ,

  $$\displaystyle \begin{aligned} D_T(\varOmega,\varepsilon) \subset \lambda D_T(\varOmega,\varepsilon) = D_T(\lambda\varOmega, \lambda^n\varepsilon)\subset D_T(\lambda\varOmega, \varepsilon). \end{aligned}$$

• 2. If Ω is symmetric about the origin, then $\mathcal {H}_\varOmega ^\varepsilon$ is symmetric about the origin as well.

• 3. Assume Ω is symmetric about the origin, let λ →∞, the secondary power diagram of $\overline {\mathcal {H}}_{\lambda \varOmega }^\varepsilon$ forms a complete fan of $\mathbb {R}{ }^k$.

• 4. The projection of the upper and lower envelopes Env({π _T, T ∈ Σ(Y )}) to $\mathbb {R}{ }^k$ (treated as the secondary power diagram of $\mathbb {R}{ }^k$ ) is a complete fan.

Proof

• 1. From the secondary power diagram theorems, it is obvious that if h ∈ D _T(Ω, ε) then ∀0 < μ ≤ 1, μh ∈ D _T(Ω, ε). Also, for any convex set $P\subset \mathbb {R}{ }^n$, vol(λP) = λ ⁿ vol(P). Hence the equations hold.

• 2. Assume Ω is symmetric about the origin, T is a coherent triangulation T ∈ Σ(Y ). If D ⁺(h) is the nearest power diagram induced by h, the dual weighted Delaunay triangulation T ⁺(h) = T, then − h induces the furthest power diagram D ⁻(−h), which is symmetric to D ⁺(h) about the origin, the dual weighted Delaunay triangulation T ⁻(−h) = T as well. Hence $-h\in \mathcal {H}_\varOmega ^\varepsilon$ as well.

• 3. For all λ > 0, the upper envelope of the secondary power diagram U(Y ) =max_{T ∈ Σ(Y )} π _T remains the same, independent of λ. When λ goes to infinity, each cell becomes an infinite cone. The apexes of all the cones are at the origin, hence the limit of the secondary power diagrams form a complete fan of $\mathbb {R}{ }^k$.

• 4. The projection of the upper and lower envelopes Env({π _T, T ∈ Σ(Y )}) to $\mathbb {R}{ }^k$ is independent of the choice of Ω and ε, from 3 it is limit of a secondary power diagram $\mathcal {H}_\varOmega ^\varepsilon$ for a central symmetric Ω. So it is a complete fan.

1.3 I Proof of Theorem 23 Transformations Among Triangulations

Corollary 26 (Transformations Among Triangulations)

Given a point configuration $Y=\{y_1,y_2,\dots ,y_k\}\subset \mathbb {R}{ }^n$ , T ₀, T ₁ ∈ Σ(Y ) are two non-degenerated coherent triangulations, then one can transform T ₀ to T ₁ by a sequence of bistellar transformations, such that all the intermediate triangulations are non-degenerated.

Proof

Because T ₀ and T ₁ are non-degenerated coherent triangulations, their corresponding height vectors h ₀ and h ₁ belong to the Alexandrov power diagram space $\mathcal {H}_\varOmega ^0(Y)$. Due to the convexity of $\mathcal {H}_\varOmega ^0(Y)$, the line segment γ(t) = (1 − t) h ₀ + t h ₁, t ∈ [0, 1] is inside $\mathcal {H}_\varOmega ^0(Y)$. γ(t) won’t touch the boundary of $\mathcal {H}_\varOmega ^0(Y)$, therefore, all points in γ(t) are interior points. γ(t) crosses a finite number of cells, each cell corresponds to a non-degenerated coherent triangulation. When γ(t) transits from one cell to the neighbor, the triangulations is updated by a bistellar transformation.

1.4 I Example of Secondary Power Diagram

Given a point configuration $Y=\{1,0,-1\}\subset {\mathbb {R}{}}$, Ω = [−1, 1]. Construct three lines

$$\displaystyle \begin{aligned} \begin{cases} l_1 \colon & y=x-h_1,\\ l_2 \colon & y=h_1+h_2,\\ l_3 \colon & y=-x-h_2. \end{cases} \end{aligned}$$

The upper envelope and the lower envelop of the three lines are shown in Fig. 8.

Fig. 8

The upper and lower envelopes of l ₁, l ₂ and l ₃. (a) Upper envelope. (b) Lower envelope

The secondary power diagram Π(Y ) of the point configuration Y  is shown in Fig. 9, where the blue part corresponds to the upper envelop and the black part corresponds to the lower envelope.

Fig. 9

The secondary power diagram of Y