When Convexity Helps Collapsing Complexes

This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We ﬁrst consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a ﬁnite point set is collapsible. We then consider a convex domain deﬁned as the convex hull of a ﬁnite point set. We show that if the point set samples suﬃciently densely the domain, then both the Čech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a ﬁltration by sweeping space with a growing sphere whose center has been ﬁxed and studying events occurring through the ﬁltration. Since the ﬁltration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory.


Introduction
Contractibility and collapsibility.In the realm of point set topology in Euclidean spaces, a set is said to be contractible if it has the homotopy type of a point.Examples of contractible sets are bounded convex sets which can each be continuously retracted to a point.Consider a finite collection of convex sets C and the domain defined by their union C. We know that each convex set in the collection is contractible.Moreover, any non-empty intersection of two or more convex sets being itself convex is again contractible.In this situation, the topology of the domain C is determined by the pattern in which convex sets in C intersect.This is asserted by the Nerve Lemma [6] also known as Leray's Theorem.Recall that the nerve of a collection of sets is the abstract simplicial complex whose vertices correspond to sets in the collection and whose simplices correspond to sub-collections with non-empty intersection.
The nerve provides a way to record the intersection pattern of sets in a collection.The Nerve Lemma implies that the nerve inherits the homotopy type of C. In particular, if C is convex, then it is contractible and so is the nerve.
Simplicial collapses are unitary operations on simplicial complexes that preserve the homotopy type while removing a few simplices.A simplicial complex is said to be collapsible if it can be reduced to a single vertex by a sequence of collapses.Collapsibility can be interpreted as a combinatorial version of contractibility.Indeed, each sequence of collapses 11:2 When Convexity Helps Collapsing Complexes can be interpreted as a combinatorial analog of a deformation retract.Whereas contractibility involves continuous processes, the notion of collapsibility, thanks to its combinatorial and finitary nature, can be handled directly in a computational context [4,2,3].
If a simplicial complex is collapsible, then it is also contractible.However, the converse does not hold -the triangulated Bing's house is a famous example of simplicial complex which is contractible without being collapsible [9].A natural question to ask is: Under which condition a simplicial complex is collapsible?This question has been studied in different context.For instance, it is known that all triangulation of the 2-dimensional ball are collapsible.Whether linear subdivisions of convex d-balls, or at least their subdivision, are collapsible has been a long-standing open problem [5].

Contributions.
In this paper, we establish collapsibility of certain simplicial complexes.First, we consider a finite collection of compact convex sets whose union is convex and prove that the nerve of the collection is collapsible.This is a stronger result than what we obtain when applying the Nerve Lemma which only entails contractibility of the nerve.Then, we focus on two simplicial complexes built upon a finite point set S. The first one is the Čech complex C(S, r) of S with scale parameter r defined as the nerve of the collection of balls of radius r centered at S. The second one is the Vietoris-Rips R(S, r) which is the largest simplicial complex sharing with C(S, r) the same set of vertices and edges.In other words, a simplex belongs to R(S, r) if and only if all its vertices and edges belong to C(S, r).Such a simplicial complex which enjoys the property to be completely determined by the set of its vertices and edges is called a flag completion.We study the situation in which the point set S samples sufficiently densely a given convex domain.As a second result, we obtain that, in this particular case, both the Čech complex and the Vietoris-Rips complex of S are collapsible for a well-chosen scale parameter.Furthermore, when reducing the Vietoris-Rips complex to a point by a sequence of collapses, we can do it in such a way that we preserve at all time the property of the complex to be a flag completion.This opens the possibility to compute the sequence of collapses by maintaining the graph formed by the vertices and edges, thus avoiding the complexity, possibly exponential in the dimension, required by an extensive complex representation; see [1] for an example of a data structure optimized for "almost-flag" completions.

Preliminaries
In this section we review the necessary background and explain some of our terms.

Euclidean space, distances and convex sets
R d denotes the Euclidean space and x − y the Euclidean distance between two points x and y of R d .Given a point o ∈ R d and a subset X ⊆ R d , we write d(o, X) for the infimum of Euclidean distances between o and points in X.By convention, we set d(o, X) = +∞ whenever X is empty.We write ∂X for the boundary of X.The closed ball with center x and radius r is denoted by B(x, r).The dilation of X by a ball of radius r centered at the origin is X ⊕r = x∈X B(x, r) and referred to as the r-offset of X.If X is either compact or convex, so is X ⊕r .It is easy to check that (X ∪ Y ) ⊕r = X ⊕r ∪ Y ⊕r .In this paper, we will use many times the following fact.

Collapses
Let π : Vert(K) → R n be an injective map that sends the n vertices of K to n affinely independent points of R n , such as for instance the n vectors of the standard basis of R n .Let Hull(X) denote the convex hull of X ⊆ R n .The underlying space of K is the point set |K| = σ∈K Hull(π(σ)) and is defined up to a homeomorphism.We shall say that an operation preserves the homotopy-type of K if the result is a simplicial complex K whose underlying space is homotopy equivalent to that of K. We are interested in simplifying a simplicial complex through a sequence of homotopy-preserving operations.
Consider the operation that removes from K the set of simplices ∆ = St K (σ).This operation is known to preserve the homotopy-type in the following three cases: 1. ∆ = {σ, τ } with σ = τ .This case can also be characterized by the fact that the link of σ is reduced to a singleton.The operation is called an (elementary) collapse.2. ∆ = {η | σ ⊆ η ⊆ τ } with σ = τ .This case can also be characterized by the fact that the link of σ is the closure of a simplex.The operation is called a (classical) collapse.3. The link of σ is a cone.The operation is called an (extended) collapse.Both classical and extended collapses can be expressed as compositions of elementary collapses.A simplicial complex is said to be collapsible if it can be reduced to a single vertex by a finite sequence of collapses (either elementary, classical or extended).

Filtrations
In the next two sections, we establish collapsibility of certain simplicial complexes using the following strategy.We associate to simplicial complex K a filtration {K(t)} t∈R which is a nested one-parameter family of simplicial complexes such that K(−∞) = ∅ and K(+∞) = K.The filtration induces a strict order on simplices of K defined by η ≺ K ν ⇔ {η ∈ K(t i ) and ν ∈ K(t j ) \ K(t i ) for some t i < t j }.Simply put, as time t goes by, if a simplex η shows up in K(t) strictly before another simplex ν, then η ≺ K ν.As we continuously increases the parameter t from −∞ to +∞, the simplicial complex K(t) changes only at finitely many values of t for which the set of simplices ∆(t) = K(t) \ lim u→t − K(u) is non-empty, where lim u→t − K(u) designates the limit of K(u) as u approaches t from below.Let t 0 be the first time at which a non-empty simplicial complex appears in the filtration.In other words, K(t 0 ) S o C G 2 0 1 9 11:4 When Convexity Helps Collapsing Complexes is the smallest non-empty simplicial complex in the filtration.To prove that K is collapsible, it suffices to show that: (1) K(t 0 ) is collapsible; (2) the operation that removes ∆(t) from K(t) is a collapse for all t > t 0 .For this purpose, it will be crucial to build filtrations that are simple (see Figure 1): Definition 1.The filtration {K(t)} t∈R is simple if for all t > t 0 , then ∆(t) has a unique inclusion-minimal element σ t .
Figure 1 The 6 simplicial complexes form a filtration of K which is simple.
When the filtration is simple, ∆(t) is precisely the star of σ t in K(t) and removing ∆(t) from K(t) is a collapse if one of the three conditions listed in Section 2.3 is satisfied.
Definition 2. Two simplices σ 1 and σ 2 are in conjunction in filtration {K(t)} t∈R if they are both inclusion-minimal elements of ∆(t) for some t > t 0 (see Figure 2).Clearly, a filtration is simple if and only if it contains no pair of simplices in conjunction.
The following definition will be useful when, in the next section, we perturb a filtration to make it simple.Definition 3. Consider a filtration {K(t)} t∈R of K and a filtration {L(t)} t∈R of L. We say that the filtration {L(t)} t∈R is finer than the filtration

3
Collapsing nerves of compact convex sets

Statement of results
Given a finite collection of sets C, we write C for the union of sets in C and C for the common intersection of sets in C. The nerve of C is the abstract simplicial complex that consists of all non-empty subcollections whose sets have a non-empty common intersection, The nerve theorem implies that if all sets in C are compact and convex, then the nerve of C is homotopy equivalent to the union of sets in C, that is Nrv C C. We get immediately that if furthermore C is a non-empty convex set, then Nrv C is contractible.In this paper, we prove that under the same hypotheses on C, we have a stronger result, namely, that Nrv C is collapsible.Formally: Two corollaries follow immediately.First, we obtain collapsibility of the Delaunay complex.Recall that given a finite point set Before proving the corollary, notice that the condition on S is tight as for any δ > 0, if Hull(S) ⊆ S ⊕(r+δ) , then C(S, r) is not necessarily collapsible.Take for instance the Cech complex of two points at distance 2(r + δ).
Proof.Apply Theorem 4 to the collection C = {B(s, r) ∩ Hull(S) | s ∈ S} and notice that Nrv C is isomorphic to C(S, r).
We prove Theorem 4 by adopting the following strategy.Consider a finite collection C of compact convex sets whose union C is non-empty and convex.We first build a filtration of Nrv C from which we derive a sequence of collapses reducing Nrv C to a vertex.The rest of the section is devoted to the proof of Theorem 4. In Section 3.2, we build the filtration and show that the smallest non-empty simplicial complex in the filtration is collapsible.In Section 3.3, we show that we can always perturb the collection C so that the filtration associated to Nrv C enjoys nice properties (in a sense to be made precise).In Section 3.4, we study the filtration of the perturbed collection and show that events through the filtration are collapses.

Building a filtration
Let C be a family of subsets of R d whose union C is non-empty.Let o be a fix point in R d that belongs to C. We build a filtration of Nrv C by sweeping the space R d with a sphere centered at o and whose radius t ≥ 0 continuously increases from 0 to +∞.Simplicial complexes K o,C (t) in the filtration are obtained by keeping simplices in Nrv C which are subcollections of C whose common intersections have a distance to o equal to or less than t: consists of τ 0 together with all its faces and is clearly collapsible.

Perturbing filtrations
Let C be a finite family of compact convex sets whose union C is convex and non-empty and pick a point o in C. In this section, we establish that it is always possible to perturb the family C into a family of compact convex sets so as to make the filtration K o,C (t) simple while leaving C convex and Nrv C unchanged.Roughly speaking, we shall perturb sets in the collection by thickening them.Lemma 10.Let X be a finite collection of compact sets.There exists ε > 0 such that for all non-negative maps ξ : X → R bounded above by ε, we have (Nrv X) ⊕ξ = Nrv(X ⊕ξ ).
Lemma 11.There exists ε > 0 such that for all non-negative maps α : C → R bounded above by ε and all 0 Proof.Consider a non-negative map α bounded above by ε and 0 To prove (1), we apply Lemma 10 to the set X = C ∪ { C} and the map ξ defined by ξ(C) = α(C) for all C ∈ C and ξ( C) = β.We know that for ε > 0 small enough, Nrv X = Nrv(X ⊕ξ ) or equivalently η = ∅ ⇔ f α,β (η) = ∅ for all η ⊆ C, showing that (1) holds.To prove (2), write and let us establish that L α,β (t) is finer than K(t).As we continuously increases the parameter t from −∞ to +∞, the simplicial complex K(t) changes only at finitely many values 0 = t 0 < . . .< t m of t.Let B i = B(o, t i ) for i ∈ {0, . . ., m}.Let us apply Lemma 10 to the set X = C ∪ { C} ∪ {B 0 , . . ., B m } and the map ξ defined by ξ(C) = α(C) for all C ∈ C, ξ( C) = β and ξ(B i ) = 0. Lemma 10 implies that for ε > 0 small enough, we have the following five equivalences: Thus, for all t ≥ 0, there exists i such that Next lemma ensures that when we replace one of the set C in C by C ⊕ε ∩ C, some of the events in the corresponding filtration happen at an earlier time.Precisely: As we go from x to x on the segment connecting x to x , the distance to o decreases in a sufficiently small neighborhood of x while we remain in both sets C ⊕ε and η ∩ C. Thus, Before proving Lemma 9, recall that two simplices σ 1 and σ 2 are in conjunction in filtration K o,C (t) if they are both inclusion-minimal elements of ∆ o,C (t) for some t > 0.

S o C G 2 0 1 9 11:8 When Convexity Helps Collapsing Complexes
Proof of Lemma 9.The proof consists in applying a sequence of elementary perturbations to set C while preserving Nrv C and C until no two simplices remain in conjunction in the filtration K o,C (t).Suppose two simplices are in conjunction, say σ 1 and σ 2 .Suppose C ∈ σ 1 and C ∈ σ 2 and replace the convex set C by C ⊕ε ∩ C. Clearly, C is still a collection of compact convex sets, C is left unchanged by the operation and Lemma 11 implies that for ε > 0 small enough, the nerve of C is also left unchanged.We prove below that for ε > 0 small enough: (1) σ 1 and σ 2 are not in conjunction anymore after the operation; (2) two simplices η and ν are not in conjunction after the operation unless a face η ⊆ η and a face ν ⊆ ν were already in conjunction before the operation.Introduce the map f : (1) Let us prove that for ε > 0 small enough, f (σ 1 ) and f (σ 2 ) are not in conjunction in (2) Let us prove that for ε > 0 small enough, two simplices f (η) and f (ν) cannot be in conjunction in The result hence follows.
To remove all pair of simplices in conjunction, consider the partial order ≺ on pair of simplices defined by (ν , η ) ≺ (ν, η) if dim ν ≤ dim ν and dim η ≤ dim η.Sort all pair of simplices in conjunction according to a total order compatible with this partial order.Take the smallest pair (σ 1 , σ 2 ) and apply the elementary perturbation described above.Notice that the operation does not create any new pair of simplices in conjunction smaller than (σ 1 , σ 2 ).By repeating this operation a finite number of times, we thus get a new collection of compact convex sets possessing the same nerve and the same union but for which no two simplices are in conjunction anymore.In other words, the filtration associated to the new collection is simple.Since elementary perturbations can be made as small as wanted, their composition can be made smaller than any given δ > 0.

Studying events in the filtration
Throughout this section, C designates a finite collection of compact convex sets whose union is a non-empty convex set and o designates a point in C. In this section, we establish Theorem 4. Let us start with a few general observations.Consider t > 0 such that ∆ o,C (t) = ∅.If we assume {K o,C (t)} t∈R to be simple, then by definition ∆ o,C (t) has a unique inclusion-minimal element σ t .Let p t be the point in σ t closest to o and let Proof.Let us prove that for all η ⊆ C, we have the equivalence Consider first η such that σ t ⊆ η ⊆ τ t .We have the following sequence of inclusions showing that d(o, η) = t.Conversely, consider η ⊆ C such that d(o, η) = t and let x be the point of η closest to o. Observe that σ t ⊆ η because η ∈ ∆ o,C (t) and therefore x ∈ η ⊆ σ t .It follows that σ t contains two points x and p t such that x − o = p t − o = t = d(o, σ t ).Because σ t is convex, there is a unique point in σ t at which the smallest distance to o is achieved, showing that p t = x and therefore η ⊆ τ t .
Hence, ∆ o,C (t) has a unique inclusion-minimal element σ t and a unique inclusion-maximal element τ t .More precisely, ∆ o,C (t) consists of all cofaces of σ t in K o,C (t) and these cofaces are all faces of τ t .To prove that removing ∆ o,C (t) from K o,C (t) is a collapse, it suffices to establish that τ t = σ t .Lemma 14 below shows that p t lies on the boundary of all the convex sets in σ t .If we were able to prove that p t belongs to the interior of one of the convex sets of τ t , we would be done because we would be sure that τ t = σ t .Unfortunately, this is not true in general (see point x ACD in Figure 3 for a counterexample) but becomes true if we slightly perturb C, as explained in Lemma 15.
Suppose furthermore that all trigger points of Nrv f (C) lie in the interior of f (C).Let y be one of those trigger points.If y = o, then y lies in the interior of some f (C) ∈ f (C).
Suppose for a contradiction that y = o and that y lies on the boundary of f (C) for all C ∈ τ ; see Figure 4, right.Since Nrv f (C) = f (Nrv C), we have that y ∈ f (τ ) = ∅ implies that τ = ∅.Let x be a point in the latter intersection.For all C ∈ τ , we have that x belongs to the interior of f (C) while y belongs to the boundary of f (C).Thus, the vector y − x points outward f (C) at y for all C ∈ τ .Since all convex sets f (C ) for which C not in τ are at some positive distance from y, it follows that, on the Then, all trigger points of Nrv f (C) lie in the interior of f (C).2) the filtration {K o,f (C) (t)} t≥0 is simple.Furthermore, the map α can be chosen arbitrarily small and in particular bounded above by ( √ 2 − 1)ε.Observe that such an f satisfies the assumptions of Lemma 16 and therefore of Lemma 15.Letting D = f (C), we prove in the following paragraph that Nrv D is collapsible which entails immediately that Nrv C is collapsible as the two are isomorphic.

Collapsing Rips complexes
In this section, we turn our attention to Rips complexes.Given a point set S and a scale parameter r, the Rips complex R(S, r) is the simplicial complex whose simplices are subsets of points in S with diameter at most 2r.Rips complexes are examples of flag completions.
Recall that the flag completion of a graph G, denoted Flag(G), is the maximal simplicial complex whose 1-skeleton is G.Let G(S, r) denote the graph whose vertices are the points S and whose edges connect all pairs of points within distance 2r.The Rips complex of S with parameter r is R(S, r) = Flag(G(S, r)).It is the largest simplicial complex sharing with the Čech complex C(S, r) the same 1-skeleton.However, the Rips complex has the computational advantage over the Čech complex to be a flag completion: it suffices to compute its 1-skeleton to encode the whole complex.In this section, we prove that if S samples sufficiently densely Hull(S), then the Rips complex is collapsible for a suitable value of the scale parameter.Proof.Set r = 1 and write B x for the closed unit ball centered at x. Fix a point o in the convex hull of S. We construct a sequence of collapses by sweeping the space with a sphere centered at o and whose radius t ≥ 0 continuously decreases from +∞ to 0. Specifically, let G t be the graph whose vertices are points s ∈ S such that B(o, t) ∩ B s = ∅ and whose edges connect all pair of points a Clearly, G +∞ = G(S, 1) and K +∞ = Flag G +∞ = R(S, 1).We claim that K 0 is collapsible.Indeed, the vertex set of K 0 is the set of points τ 0 = {s ∈ S | o ∈ B s } = S ∩ B o which is non-empty since o ∈ Hull(S) ⊆ S ⊕1 .It follows that K 0 = Flag G 0 consists of τ 0 and all S o C G 2 0 1 9 11:12 When Convexity Helps Collapsing Complexes its faces and is collapsible.As we continuously decreases t from +∞ to 0, changes in the simplicial complex K t occur whenever a vertex or an edge disappears from the graph G t ; see Figure 6.Generically, we may assume that these events do not happen simultaneously.
When a vertex a disappears from G t at time t, the intersection B(o, t) ∩ B a reduces to a single point x; see Figure 7, left.In this situation, we claim that the link of a in K t is the closure of the simplex τ x = S ∩ B x \ {a}.First, note that τ x is non-empty since x lies on the segment connecting o to a and therefore belongs to the convex hull of S which is contained in S ⊕2− √ 3 .Hence, there is a point s ∈ S in the interior of B x and τ x = ∅.Furthermore, τ x is precisely the vertex set of the link since an edge au belongs to K t if and only if B(o, t) ∩ B a ∩ B u = ∅ with u ∈ S \ {a} which can be reformulated as u ∈ S ∩ B x \ {a}.Finally, any two vertices u and v in the link are connected by an edge since clearly u, v ∈ τ x implies B(o, t) ∩ B u ∩ B v ⊃ {x}.We have just proved that the link of a in K t is the closure of a simplex.Thus, removing the star of a from K t is a classical collapse.showing that uv also belongs to the link of ab in K t .We have just proved that the link of ab in K t is a cone.Thus, removing the star of ab from K t is an extended collapse.

Two geometric lemmas
The proof of Theorem 17 relies on two geometric lemmas.The first one states facts about three disks in the plane that intersect pairwise but have no common intersection (Lemma 18).It will allow us to deduce facts about the way four balls intersect in R d (Lemma 19).As before, B x denotes the unit closed ball centered at x. .We claim that q c belongs to the convex hull of α, β and q ab .Indeed, for x ∈ {a, b, c}, let H x be the half-plane that contains D x and avoids the interior of D m .We have α ∈ H a ∩ H c , β ∈ H b ∩ H c , and q ab ∈ H a ∩ H b .The triangle αβq ab covers the closure of R 2 \ (H a ∪ H b ∪ H c ) and therefore q c .Let us prove that q c − q ab ≤ √ 3 − 1.For x ∈ {a, b, c}, we denote the center of D x by z x and its radius by ρ x .We are going to transform the three disks D a , D b and D c in such a way that after the transformation: (i) the three disks intersect pairwise but have no common intersection; (ii) the distance between q c and q ab is at least as large as it was before the transformation; (iii) ρ x ≤ 1 for x ∈ {a, b, c}; (iv) the centers z a , z b , and z c form an equilateral triangle of side length two.Let q c be the point on the boundary of D m that is farthest away from q ab ; see Figure 8, right.Clearly, q c − q ab ≥ q c − q ab .The two tangency points q a and q b decompose the boundary of D m in two arcs and it is not difficult to see that one of them contains both S o C G 2 0 1 9

Figure 2
Figure 2The 5 simplicial complexes form a filtration of K which is not simple as simplices CD and E are in conjunction.
∀s ∈ S} and the Delaunay complex of S is the nerve of Voronoi regions, Del(S) = Nrv{V s | s ∈ S}.Corollary 5.The Delaunay complex of any finite point set S ⊆ R d is collapsible.Proof.Apply Theorem 4 to the collection C = {V s ∩ B | s ∈ S}, where B is any ball large enough to intersect all common intersection s∈σ V s for σ ranging over Nrv S. The result follows because Del(S) is isomorphic to Nrv C which is collapsible by Theorem 4. To state the second corollary, recall that the Čech complex of a finite point set S ⊆ R d with parameter r ∈ R is the nerve of the collection of balls, C(S, r) = Nrv{B(s, r) | s ∈ S} and denote by Hull(S) the convex hull of S. Corollary 6.The Čech complex of a finite point set S ⊆ R d with parameter r ∈ R is collapsible whenever Hull(S) ⊆ S ⊕r .
Clearly, K o,C (t) = ∅ for all t < 0 and K o,C (+∞) = Nrv C. Notice that K o,C (0) is non-empty because point o belongs to at least one set C in the collection C and thus K o,C (0) contains at least vertex C. It follows that K o,C (0) is the smallest non-empty simplicial complex in the filtration.As we continuously increases the parameter t from −∞ to +∞, the simplicial complex K o,C (t) changes only at finitely many values of t for which the set of simplices∆ o,C (t) = {η ⊆ C | d(o, η) = t}is non-empty.When these events happen, the sphere centered at o with radius t passes through particular points of C that we call trigger points and defined below; see Figure3.Definition 7. Consider η ⊆ C such that η = ∅.The trigger point of η (with respect to o) is the point of η at which the smallest distance to o is achieved.There is thus one trigger point per simplex in Nrv C and the set of all these points is referred to as the trigger points of Nrv C (with respect to o).

Figure 3
Figure 3Left: Collection of five convex sets {A, B, C, D, E} whose union is convex.The nerve possesses six trigger points (red dots) among which one of them xAB lies on the boundary of the union and another one xACD does not lie in the interior of any convex set in the collection.The filtration associated to the nerve is depicted in Figure2and is not simple.Right: as we offset sets in the collection while keeping the union convex and the nerve unchanged, the trigger point xAB moves in the interior of the union and the trigger point xACD moves in the interior of at least one convex set (namely A).The associated filtration is depicted in Figure1and is simple.

Lemma 9 .
For all δ > 0, there exists a non-negative map α : C → R bounded above by δ such that if we replace each set C ∈ C by set C ⊕α(C) ∩ C, we perturb C in such a way that (1) the nerve of C is left unchanged; (2) the filtration K o,C (t) becomes simple.The proof of Lemma 9 relies on three technical lemmas.To state them, we need some notation and definitions.A perturbation of C is a map f : C → P(R d ), where P(R d ) designates the power set of R d .For a subcollection η ⊆ C and a simplicial complex L over C, we writef (η) = {f (C) | C ∈ η} and f (L) = {f (η) | η ∈ L}.For any non-negative map α : C → R, we write η ⊕α = {C ⊕α(C) | C ∈ η} and L ⊕α = {η ⊕α | η ∈ L}.A simple compactness argument implies the first lemma of which the proof is omitted.
then η is a proper subset of {C}∪η and the minimality of {C} ∪ η in ∆ o,C (t) implies that d(o, η) < t and therefore d(o, η ∩ C) < t.If η = ∅, observe that the later inequality holds because we have assumed that d(o, C) < t.Let x be the point in C ∩ η closest to o and let x be the point in η ∩ C closest to o.

Lemma 14 .
Consider η ∈ Nrv C and let x be the point in η closest to o. Suppose o = x.If x lies in the interior of some C, then x is also the point in (η \ {C}) closest to o. Proof.Suppose that x lies in the interior of some C ∈ η; see Figure 4, left.Because o = x, we cannot have η = {C} and therefore η = η \ {C} is non-empty.Let us prove that x is the point of η closest to o. Suppose for a contradiction that there exists a point x in η closer to o than x.Since the map m → m − o is convex and x − o > x − o , the distance to o would be decreasing along the segment [x, x ] in the vicinity of x and since this segment, in the vicinity of x, is contained in η this would contradict the fact that x is the closest point to o in η.

Figure 4
Figure 4 Left: Counterexample to Lemma 14 when sets in C are not convex.Right: Notation for the proof of Lemma 15.
segment starting from y in the direction y − x, points sufficiently close to y do not belong to f (C).In other words, y ∈ ∂ f (C).But, y being a trigger point, y lies in the interior of f (C), yielding a contradiction.Next lemma gives an example of perturbation f of C which ensures that after perturbing C with f , all trigger points of Nrv f (C) are in the interior of f (C); see Figure3.Lemma 16.Consider ε > 0 and a map α :

Figure 5
Figure 5 Notation for the proof of Lemma 16.

Figure 6
Figure 6From left to right: Our proof technique (illustrated here when S = {a, b, c, d}) consists in sweeping space with a sphere centered at o ∈ Hull(S) and whose radius t continuously decreases from +∞ to 0. We deduce from the sweep a sequence of collapses reducing R(S, α) to a vertex.

Figure 7
Figure 7Notation for the proof of Theorem 17.Two kinds of events may occur: either a vertex collapse (on the left) or an edge collapse (on the right).The edge collapse is illustrated when triangle oab is equilateral.

Figure 8
Figure 8 Notation for the proof of Lemma 18.

Lemma 18 .
Let D a , D b and D c be three disks with radius equal to or less than one and such that any two disks have a non-empty intersection while the three together have no common intersection.Let q ab be the point of D a ∩ D b closest to the center of D c .There exists a point q c ∈ D c such that: for all points α ∈ D a ∩ D c and β ∈ D b ∩ D c , the point q c is in the convex hull of α, β and q ab ; q c − q ab ≤ √ 3 − 1. Proof.Consider the disk D m whose boundary is tangent to the boundaries of the three disks D a , D b and D c and whose interior intersects none of the three disks D a , D b and D c .For x ∈ {a, b, c}, the two disks D x and D m intersect in a single point q x ; see Figure 8, left.Let α ∈ D a ∩ D c and β ∈ D b ∩ D c By Lemma 8, K o,D (0) is collapsible.Let us prove that for all t > 0 such that ∆ o,D (t) = ∅, the operation that removes ∆ o,D (t) from K o,D (t) is a collapse.Since the filtration {K o,D (t)} t≥0 is simple, ∆ o,D (t) has a unique inclusion-minimal element σ t .Let p t be the point in σ t closest to o and let τ t = {D ∈ D | p t ∈ D}.By Lemma 13, ∆ o,D (t) is the set of simplices η ∈ Nrv D such that σ t ⊆ η ⊆ τ t .Let us show that σ t = τ t .By Lemma 14, p t lies on the boundary of all sets in σ t .By Lemma 15, p t lies in the interior of at least one set D in τ t .Hence, τ t = σ t and the operation that removes ∆ o,D (t) from K o,D (t) is a collapse.