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.
Keywords
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Appendix
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 \(Conv(\{\pi _T^*\})\),
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 ) =maxT ∈ 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
The upper envelope and the lower envelop of the three lines are shown in Fig. 8.
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.
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Lei, N., Chen, W., Luo, Z., Si, H., Gu, X. (2019). Secondary Power Diagram, Dual of Secondary Polytope. In: Garanzha, V., Kamenski, L., Si, H. (eds) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol 131. Springer, Cham. https://doi.org/10.1007/978-3-030-23436-2_1
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DOI: https://doi.org/10.1007/978-3-030-23436-2_1
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