In this short paper, we consider the functional density on sets of uniformly
bounded triangulations with fixed sets of vertices. We prove that if a
functional attains its minimum on the Delaunay triangulation, for every finite
set in the plane, then for infinite sets the density of this functional attains
its minimum also on the Delaunay triangulations

Dolbilin, Nikolay P.

Edelsbrunner, Herbert

Musin, Oleg R.

English

arXiv

Dolbilin, Nikolai

Musin, Oleg

IST Austria: PubRep (Institute of Science and Technology)

On the optimality of functionals over triangulations of Delaunay sets

Nikolai P Dolbilin

Herbert Edelsbrunner

Oleg R Musin

Crossref

Russian Mathematical Surveys

Edelsbrunner, Herbert 

ISTA Research Explorer (Institute of Science and Technology Austria)

Communications of the Moscow Mathematical Society 781On the optimality of functionals over triangulations of Delaunay setsN.P. Dolbilin, H. Edelsbrunner, and O.R. MusinIn this short paper we consider the functional density on sets of uniformly boundedtriangulations with fixed sets of vertices. We prove that if a functional attains its minimumon the Delaunay triangulation for every finite set in the plane, then for infinite sets thedensity of this functional attains its minimum also on the Delaunay triangulations.A Delaunay set in Ed is a set of points X for which there are positive numbers r and Rsuch that every open d-ball of radius r contains at most one point and every closed d-ballof radius R contains at least one point of X. In this paper we consider Delaunay sets ingeneral position, that is, no d+ 2 points in X lie on a common (d− 1)-sphere.By a triangulation of X we mean a simplicial complex whose vertex set is X. Forfinite sets the simplices decompose the convex hull of the set, while for Delaunay sets Xthe simplices decompose Ed. We say that a triangulation T is uniformly bounded if thereexists a positive number q = q(T ) that is greater than or equal to the circumradii of alld-simplices in the triangulation: R(S) 6 q for all d-simplices S of T . We denote the familyof all uniformly bounded triangulations of X by Θ(X).Delaunay sets were introduced by BorisDelaunay (1924), who called them (r,R)-systems.He proved that for any Delaunay set X there exists a unique Delaunay tesselation DT (X)(see, for instance, [1]). If X is in general position, then DT (X) is a triangulation of X inthe sense defined above. Since the circumradius of any simplex is at most R, the Delaunaytriangulation is uniformly bounded with q = R, that is, DT (X) ∈ Θ(X). We note thatevery Delaunay set also has triangulations that are not uniformly bounded, and it is notdifficult to construct them.We want to remind the reader of a related open problem about Delaunay sets: is it truethat for every planar Delaunay set X and every positive number C there exists a triangle∆ that contains none of the points in X and has area greater than C? While we heard ofthis question from Michael Boshernitzan, it is sometimes referred to as Danzer’s problem.Let F be a functional defined on d-simplices S. (For instance, F (S) may be the sum ofsquares of edge lengths multiplied by the volume of S.) We only consider functionals thatare continuous with respect to the parameters describing the simplices, for example, thelengths of their edges. Let X be a finite set in Ed and T any triangulation of X. Then Fcan be defined on T as F (T ) =∑S∈T F (S).It is clear that this definition cannot be used for infinite sets. We therefore define the(lower) density of F for a uniformly bounded triangulation T of a Delaunay set X asF (T ) := limα→∞1vol(Bα)∑S⊂BαF (S),where Bα denotes the closed ball of radius α with centre at the origin of Ed. For the restof the paper, we limit ourselves to dimension d = 2.Theorem. Let F be a continuous functional that attains its minimum for every finite setY ⊂ E2 for the Delaunay triangulation of Y . Then the density F on Θ(X), where X ⊂ E2is a Delaunay set, attains its minimum for the Delaunay triangulation of X .This research was supported by the Russian Federation project no. 11.G34.31.0053, grant no.11-01-00735 from the Russian Foundation for Basic Research, and DMS-1101688.AMS 2010 Mathematics Subject Classification. Primary 52C20, 52C22.DOI 10.1070/RM2012v067n04ABEH004807.782 Communications of the Moscow Mathematical SocietyProof. Let T ∈ Θ(X) be a triangulation with parameter q and consider the simplicialcomplex τα(T ) that consists of all triangles, edges, and vertices of T contained in Bα. Weconsider the convex hull Cα of the vertices of τα(T ).The difference between Cα and the union of the triangles in τα(T ) consists of polygons,and since any polygon can be triangulated without adding vertices, τα can be extendedto a triangulation ψα of the same set of vertices.Write Kα for the number of triangles in τα(T ). Since the circumradius of each triangleis bounded from above, and the lengths of its edges are bounded from below, the areaof each triangle is at least some constant. It follows that Kα is at most some constanttimes α2. The circle bounding Bα intersects at most some constant times√Kα of thetriangles in T , which implies that ψα has at most a constant times√Kα triangles inaddition to those in τα. Using the continuity of the functional, it follows thatlimα→∞F (τα)F (ψα)= 1. (∗)By the assumption, F (ψα) is no less than the value of F on the Delaunay triangulationof the same set of vertices, which completes the proof.We remark that there are non-convex polytopes in dimension d > 2 that cannot betriangulated without adding new vertices. They constitute the main difficulty in extend-ing the theorem to general dimensions. The theorem and the results stated in thepapers [1]– [5] yield the following result.Corollary. Let ∆ be a triangle with barycentre b, circumcentre c, and edges of lengthsa1, a2, a3 . Let us consider the following functionals:1) F1(∆) := Ra(∆), where R(∆) is the circumradius and a > 0;2) F2(∆) :=a21 + a22 + a23s(∆), where s(∆) is the area of ∆;3) F3(∆) := −ρ(∆), where ρ(∆) is the inradius of ∆;4) F4(∆) := (a21 + a22 + a23)s(∆);5) F5(∆) := Ra(∆)s(∆), where a > 1;6) F6(∆) := ∥b(∆)− c(∆)∥2s(∆).Then the densities F i , i = 1, . . . , 6, attain their minima on the Delaunay triangulationsof Delaunay sets in the plane.For finite sets the optimality of the functionals F1 and F2 was shown in [3], the opti-mality of F3 was shown in [2], the optimality of F4 was shown [5], and the optimality ofF5 and F6 was shown in [4].Bibliography[1] Б. Н. Делоне, УМН, 1937, no. 3, 16–62. [B.N. Delaunay, Uspekhi Mat. Nauk, 1937,no. 3, 16–62.][2] T. Lambert, Proceedings of the 6th Canadian Conference on ComputationalGeometry (Saskatoon, SK, Canada 1994), University of Saskatchewan 1994,pp. 201–206.[3] O. R. Musin, SCG’ 97 Proceedings of the 13th Annual ACM Symposium onComputational Geometry (Nice, France 1997), ACM, New York 1997, pp. 424–426.[4] O.R. Musin, Geometry, topology, algebra and number theory, applications,International Conference dedicated to 120th anniversary of B.N. Delone, 2010,pp. 166–167, http://delone120.mi.ras.ru/app/Musin.pdf.Communications of the Moscow Mathematical Society 783[5] V. T. Rajan, Discr. Comput. Geometry 12:2 (1994), 189–202.N.P. DolbilinSteklov Mathematical Instituteof the Russian Academy of SciencesH. EdelsbrunnerInstitute of Science and Technology AustriaO.R. MusinUniversity of Texas at Brownsville, USAPresented by V.M. BuchstaberAccepted 21/JUN/12Translated by H. EDELSBRUNNERand O. MUSIN

Communications of the Moscow Mathematical Society 781On the optimality of functionals over triangulations of Delaunay setsN.P. Dolbilin, H. Edelsbrunner, and O.R. MusinIn this short paper we consider the functional density on sets of uniformly boundedtriangulations with fixed sets of vertices. We prove that if a functional attains its minimumon the Delaunay triangulation for every finite set in the plane, then for infinite sets thedensity of this functional attains its minimum also on the Delaunay triangulations.A Delaunay set in Ed is a set of points X for which there are positive numbers r and Rsuch that every open d-ball of radius r contains at most one point and every closed d-ballof radius R contains at least one point of X. In this paper we consider Delaunay sets ingeneral position, that is, no d+ 2 points in X lie on a common (d− 1)-sphere.By a triangulation of X we mean a simplicial complex whose vertex set is X. Forfinite sets the simplices decompose the convex hull of the set, while for Delaunay sets Xthe simplices decompose Ed. We say that a triangulation T is uniformly bounded if thereexists a positive number q = q(T ) that is greater than or equal to the circumradii of alld-simplices in the triangulation: R(S) 6 q for all d-simplices S of T . We denote the familyof all uniformly bounded triangulations of X by Θ(X).Delaunay sets were introduced byBorisDelaunay (1924), who called them (r,R)-systems.He proved that for any Delaunay set X there exists a unique Delaunay tesselation DT (X)(see, for instance, [1]). If X is in general position, then DT (X) is a triangulation of X inthe sense defined above. Since the circumradius of any simplex is at most R, the Delaunaytriangulation is uniformly bounded with q = R, that is, DT (X) ∈ Θ(X). We note thatevery Delaunay set also has triangulations that are not uniformly bounded, and it is notdifficult to construct them.We want to remind the reader of a related open problem about Delaunay sets: is it truethat for every planar Delaunay set X and every positive number C there exists a triangle∆ that contains none of the points in X and has area greater than C? While we heard ofthis question from Michael Boshernitzan, it is sometimes referred to as Danzer’s problem.Let F be a functional defined on d-simplices S. (For instance, F (S) may be the sum ofsquares of edge lengths multiplied by the volume of S.) We only consider functionals thatare continuous with respect to the parameters describing the simplices, for example, thelengths of their edges. Let X be a finite set in Ed and T any triangulation of X. Then Fcan be defined on T as F (T ) =∑S∈T F (S).It is clear that this definition cannot be used for infinite sets. We therefore define the(lower) density of F for a uniformly bounded triangulation T of a Delaunay set X asF (T ) := limα→∞1vol(Bα)∑S⊂BαF (S),where Bα denotes the closed ball of radius α with centre at the origin of Ed. For the restof the paper, we limit ourselves to dimension d = 2.Theorem. Let F be a continuous functional that attains its minimum for every finite setY ⊂ E2 for the Delaunay triangulation of Y . Then the density F on Θ(X), where X ⊂ E2is a Delaunay set, attains its minimum for the Delaunay triangulation of X .This research was supported by the Russian Federation project no. 11.G34.31.0053, grant no.11-01-00735 from the Russian Foundation for Basic Research, and DMS-1101688.AMS 2010 Mathematics Subject Classification. Primary 52C20, 52C22.DOI 10.1070/RM2012v067n04ABEH004807.782 Communications of the Moscow Mathematical SocietyProof. Let T ∈ Θ(X) be a triangulation with parameter q and consider the simplicialcomplex τα(T ) that consists of all triangles, edges, and vertices of T contained in Bα. Weconsider the convex hull Cα of the vertices of τα(T ).The difference between Cα and the union of the triangles in τα(T ) consists of polygons,and since any polygon can be triangulated without adding vertices, τα can be extendedto a triangulation ψα of the same set of vertices.Write Kα for the number of triangles in τα(T ). Since the circumradius of each triangleis bounded from above, and the lengths of its edges are bounded from below, the areaof each triangle is at least some constant. It follows that Kα is at most some constanttimes α2. The circle bounding Bα intersects at most some constant times√Kα of thetriangles in T , which implies that ψα has at most a constant times√Kα triangles inaddition to those in τα. Using the continuity of the functional, it follows thatlimα→∞F (τα)F (ψα)= 1. (∗)By the assumption, F (ψα) is no less than the value of F on the Delaunay triangulationof the same set of vertices, which completes the proof.We remark that there are non-convex polytopes in dimension d > 2 that cannot betriangulated without adding new vertices. They constitute the main difficulty in extend-ing the theorem to general dimensions. The theorem and the results stated in thepapers [1]– [5] yield the following result.Corollary. Let ∆ be a triangle with barycentre b, circumcentre c, and edges of lengthsa1, a2, a3 . Let us consider the following functionals:1) F1(∆) := Ra(∆), where R(∆) is the circumradius and a > 0;2) F2(∆) :=a21 + a22 + a23s(∆), where s(∆) is the area of ∆;3) F3(∆) := −ρ(∆), where ρ(∆) is the inradius of ∆;4) F4(∆) := (a21 + a22 + a23)s(∆);5) F5(∆) := Ra(∆)s(∆), where a > 1;6) F6(∆) := ∥b(∆)− c(∆)∥2s(∆).Then the densities F i , i = 1, . . . , 6, attain their minima on the Delaunay triangulationsof Delaunay sets in the plane.For finite sets the optimality of the functionals F1 and F2 was shown in [3], the opti-mality of F3 was shown in [2], the optimality of F4 was shown [5], and the optimality ofF5 and F6 was shown in [4].Bibliography[1] Б.Н. Делоне, УМН, 1937, no. 3, 16–62. [B.N. Delaunay, Uspekhi Mat. Nauk, 1937,no. 3, 16–62.][2] T. Lambert, Proceedings of the 6th Canadian Conference on ComputationalGeometry (Saskatoon, SK, Canada 1994), University of Saskatchewan 1994,pp. 201–206.[3] O.R. Musin, SCG’ 97 Proceedings of the 13th Annual ACM Symposium onComputational Geometry (Nice, France 1997), ACM, New York 1997, pp. 424–426.[4] O.R. Musin, Geometry, topology, algebra and number theory, applications,International Conference dedicated to 120th anniversary of B.N. Delone, 2010,pp. 166–167, http://delone120.mi.ras.ru/app/Musin.pdf.Communications of the Moscow Mathematical Society 783[5] V.T. Rajan, Discr. Comput. Geometry 12:2 (1994), 189–202.N.P. DolbilinSteklov Mathematical Instituteof the Russian Academy of SciencesH. EdelsbrunnerInstitute of Science and Technology AustriaO.R. MusinUniversity of Texas at Brownsville, USAPresented by V.M. BuchstaberAccepted 21/JUN/12Translated by H. EDELSBRUNNERand O. MUSIN

Dolbilin, Nikolai P

Musin, Oleg R

IST PubRep

Communications of the Moscow Mathematical Society 781
On the optimality of functionals over triangulations of Delaunay sets
N.P. Dolbilin, H. Edelsbrunner, and O.R. Musin
In this short paper we consider the functional density on sets of uniformly bounded
triangulations with fixed sets of vertices. We prove that if a functional attains its minimum
on the Delaunay triangulation for every finite set in the plane, then for infinite sets the
density of this functional attains its minimum also on the Delaunay triangulations.
A Delaunay set in Ed is a set of points X for which there are positive numbers r and R
such that every open d-ball of radius r contains at most one point and every closed d-ball
of radius R contains at least one point of X. In this paper we consider Delaunay sets in
general position, that is, no d+ 2 points in X lie on a common (d− 1)-sphere.
By a triangulation of X we mean a simplicial complex whose vertex set is X. For
finite sets the simplices decompose the convex hull of the set, while for Delaunay sets X
the simplices decompose Ed. We say that a triangulation T is uniformly bounded if there
exists a positive number q = q(T ) that is greater than or equal to the circumradii of all
d-simplices in the triangulation: R(S) 6 q for all d-simplices S of T . We denote the family
of all uniformly bounded triangulations of X by Θ(X).
Delaunay sets were introduced byBorisDelaunay (1924), who called them (r,R)-systems.
He proved that for any Delaunay set X there exists a unique Delaunay tesselation DT (X)
(see, for instance, [1]). If X is in general position, then DT (X) is a triangulation of X in
the sense defined above. Since the circumradius of any simplex is at most R, the Delaunay
triangulation is uniformly bounded with q = R, that is, DT (X) ∈ Θ(X). We note that
every Delaunay set also has triangulations that are not uniformly bounded, and it is not
difficult to construct them.
We want to remind the reader of a related open problem about Delaunay sets: is it true
that for every planar Delaunay set X and every positive number C there exists a triangle
∆ that contains none of the points in X and has area greater than C? While we heard of
this question from Michael Boshernitzan, it is sometimes referred to as Danzer’s problem.
Let F be a functional defined on d-simplices S. (For instance, F (S) may be the sum of
squares of edge lengths multiplied by the volume of S.) We only consider functionals that
are continuous with respect to the parameters describing the simplices, for example, the
lengths of their edges. Let X be a finite set in Ed and T any triangulation of X. Then F
can be defined on T as F (T ) =
∑
S∈T F (S).
It is clear that this definition cannot be used for infinite sets. We therefore define the
(lower) density of F for a uniformly bounded triangulation T of a Delaunay set X as
F (T ) := lim
α→∞
1
vol(Bα)
∑
S⊂Bα
F (S),
where Bα denotes the closed ball of radius α with centre at the origin of Ed. For the rest
of the paper, we limit ourselves to dimension d = 2.
Theorem. Let F be a continuous functional that attains its minimum for every finite set
Y ⊂ E2 for the Delaunay triangulation of Y . Then the density F on Θ(X), where X ⊂ E2
is a Delaunay set, attains its minimum for the Delaunay triangulation of X .
This research was supported by the Russian Federation project no. 11.G34.31.0053, grant no.
11-01-00735 from the Russian Foundation for Basic Research, and DMS-1101688.
AMS 2010 Mathematics Subject Classification. Primary 52C20, 52C22.
DOI 10.1070/RM2012v067n04ABEH004807.
782 Communications of the Moscow Mathematical Society
Proof. Let T ∈ Θ(X) be a triangulation with parameter q and consider the simplicial
complex τα(T ) that consists of all triangles, edges, and vertices of T contained in Bα. We
consider the convex hull Cα of the vertices of τα(T ).
The difference between Cα and the union of the triangles in τα(T ) consists of polygons,
and since any polygon can be triangulated without adding vertices, τα can be extended
to a triangulation ψα of the same set of vertices.
Write Kα for the number of triangles in τα(T ). Since the circumradius of each triangle
is bounded from above, and the lengths of its edges are bounded from below, the area
of each triangle is at least some constant. It follows that Kα is at most some constant
times α2. The circle bounding Bα intersects at most some constant times
√
Kα of the
triangles in T , which implies that ψα has at most a constant times
√
Kα triangles in
addition to those in τα. Using the continuity of the functional, it follows that
lim
α→∞
F (τα)
F (ψα)
= 1. (∗)
By the assumption, F (ψα) is no less than the value of F on the Delaunay triangulation
of the same set of vertices, which completes the proof.
We remark that there are non-convex polytopes in dimension d > 2 that cannot be
triangulated without adding new vertices. They constitute the main difficulty in extend-
ing the theorem to general dimensions. The theorem and the results stated in the
papers [1]– [5] yield the following result.
Corollary. Let ∆ be a triangle with barycentre b, circumcentre c, and edges of lengths
a1, a2, a3 . Let us consider the following functionals:
1) F1(∆) := Ra(∆), where R(∆) is the circumradius and a > 0;
2) F2(∆) :=
a21 + a
2
2 + a
2
3
s(∆)
, where s(∆) is the area of ∆;
3) F3(∆) := −ρ(∆), where ρ(∆) is the inradius of ∆;
4) F4(∆) := (a21 + a22 + a23)s(∆);
5) F5(∆) := Ra(∆)s(∆), where a > 1;
6) F6(∆) := ∥b(∆)− c(∆)∥2s(∆).
Then the densities F i , i = 1, . . . , 6, attain their minima on the Delaunay triangulations
of Delaunay sets in the plane.
For finite sets the optimality of the functionals F1 and F2 was shown in [3], the opti-
mality of F3 was shown in [2], the optimality of F4 was shown [5], and the optimality of
F5 and F6 was shown in [4].
Bibliography
[1] Б.Н. Делоне, УМН, 1937, no. 3, 16–62. [B.N. Delaunay, Uspekhi Mat. Nauk, 1937,
no. 3, 16–62.]
[2] T. Lambert, Proceedings of the 6th Canadian Conference on Computational
Geometry (Saskatoon, SK, Canada 1994), University of Saskatchewan 1994,
pp. 201–206.
[3] O.R. Musin, SCG’ 97 Proceedings of the 13th Annual ACM Symposium on
Computational Geometry (Nice, France 1997), ACM, New York 1997, pp. 424–426.
[4] O.R. Musin, Geometry, topology, algebra and number theory, applications,
International Conference dedicated to 120th anniversary of B.N. Delone, 2010,
pp. 166–167, http://delone120.mi.ras.ru/app/Musin.pdf.
Communications of the Moscow Mathematical Society 783
[5] V.T. Rajan, Discr. Comput. Geometry 12:2 (1994), 189–202.
N.P. Dolbilin
Steklov Mathematical Institute
of the Russian Academy of Sciences
H. Edelsbrunner
Institute of Science and Technology Austria
O.R. Musin
University of Texas at Brownsville, USA
Presented by V.M. Buchstaber
Accepted 21/JUN/12
Translated by H. EDELSBRUNNER
and O. MUSIN


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On the Optimality of Functionals over Triangulations of Delaunay Sets

On the Optimality of Functionals over Triangulations of Delaunay Sets

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