Limits of Order TypesThis work was partially supported by ANR blanc PRESAGE (ANR–11-BS02–003).

# Limits of Order Types††thanks: This work was partially supported by ANR blanc PRESAGE (ANR–11-BS02–003).

Xavier Goaoc Université de Lorraine, CNRS, INRIA, LORIA, F-54000, Nancy, France. Email: xavier.goaoc@loria.fr. This author was partially supported by Institut Universitaire de France.    Alfredo Hubard Université Paris Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEM, F-77454, Marne-la-Vallée, France. Email: alfredo.hubard@u-pem.fr.    Rémi de Joannis de Verclos Radboud University Nijmegen, Netherlands. Email: r.deverclos@math.ru.nl.    Jean-Sébastien Sereni Centre National de la Rercherche Scientifique, ICube (CSTB), Strasbourg, France. Email: sereni@kam.mff.cuni.cz.    Jan Volec Department of Mathematics, Emory University, Atlanta, USA. Email: jan@ucw.cz. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 800607. Previous affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Canada, where this author was supported by CRM-ISM fellowship.
December 3, 2018
###### Abstract

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erdős problem of finding the minimal density of 5- or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. Next we establish results on the analytic representation of limits of order types by planar measures. Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. We also show that some condition on the Hausdorff dimension of the support is necessary to obtain projective rigidity and we construct limits of order types that cannot be represented by a planar measure. Returning to combinatorial geometry we relate the regularity of this analytic representation to the aforementioned problem of Erdős on the density of -tuples in convex position, for large .

##### Keywords:

Limits of structures, flag algebra, geometric measure theory, Erdős-Szekeres theorem, Sylvester’s problem.

## 1 Introduction

The theory of dense graph limits, developed over the last decade by Borgs, Chayes, Lovász, Razborov, Sós, Szegedy, Vesztergombi and others, studies sequences of large graphs using a combination of equivalent formalisms: algebraic (as positive homomorphisms from certain graph algebras into ), analytic (as measurable, symmetric functions from  to  called graphons) and discrete probabilistic (as families  of probability distributions over -vertex graphs satisfying certain relations). These viewpoints are complementary: while the algebraic formalism allows effective computations via semi-definite methods [30], the analytic viewpoint offers powerful methods (norm equivalence, completeness) to treat in a unified setting a diversity of graph problems such as pseudorandom graphs or property testing [25].

In this article, we combine ideas from dense graph limits with order types, which are combinatorial structures arising in geometry. The order type of a point set encodes the respective positions of its elements, and suffices to determine many of its properties, for instance its convex hull, its triangulations, or which graphs admit crossing-free straight line drawings with vertices supported on that point set. Order types have received continued attention in discrete and computational geometry since the 1980s and are known to be rather intricate objects, difficult to axiomatise [33]. While order types are well defined in a variety of contexts (arbitrary dimensions, abstractly via the theory of oriented matroids) all point sets considered in this work are finite subsets of the Euclidean plane with no aligned triple, unless otherwise specified.

### 1.1 Order types and their limits

Let us first define our main objects of study.

##### Order types.

Define the orientation of a triangle  in the plane to be clockwise (CW) if  lies to the right of the line  oriented from  to  and counter-clockwise (CCW) if  lies to the left of that oriented line. (So the orientation of  is different from that of .) We say that two planar point sets  and  have the same order type if there exists a bijection  that preserves orientations: for any triple of pairwise distinct points  the triangles  and  have the same orientation. The relation of having the same order type is easily checked to be an equivalence relation; the equivalence class, for this relation, of a finite point set  is called the order type

of . A point set  with order type  is called a realization of .

When convenient, we extend to order types any notion that can be defined on a set of points and does not depend on a particular choice of realization. For instance we define the size of an order type  to be the cardinality  of any of its realization. We adopt the convention that there is exactly one order type of each of the sizes , and . We let  be the set of order types and  the set of order types of size .

##### Convergent sequences and limits of order types.

We define the density  of an order type  in another order type  as the probability that  random points chosen uniformly from a point set realizing  have order type . (Observe that this probability depends solely on the order types and not on the choice of realization.) We say that a sequence  of order types converges if the size  goes to infinity as  goes to infinity, and for any fixed order type  the sequence  of densities converges. The limit of a convergent sequence of order types  is the map

 {O→[0,1]ω↦limn→∞p(ω,ωn)

A standard compactness argument reveals that limits of order types abound. Indeed, for each element  in a sequence of order types, the map  can be seen as a point in , which is compact by Tychonoff’s theorem. Any sequence of order types with sizes going to infinity therefore contains a convergent subsequence.

### 1.2 Problems and results

We explore the application of the theory of limits of dense graphs to order types in two directions. On one hand, the algebraic description of limits as positive homomorphisms of flag algebras makes these limits amenable to semi-definite programming methods. We implemented this approach for order types and obtained numerical results. On the other hand, the fact that measures generally define limits of order types (creftype 8) unveils stimulating problems and interesting questions, of a more structural nature, on the relation between measures and limits of order types.

##### Flag algebras of order types.

The starting motivation for our work is a question raised by Erdős and Guy [18] in 1973 (see also [17]): “what is the minimum number  of convex -gons in a set of  points in the plane?”. This falls within the scope of a general (and more conceptual than precise) question of Sylvester: “what is the probability that four points at random are in convex position?”. Making sense of Sylvester’s question implies defining a distribution on -tuples of points, and from the beginning of the 20th century several variants using distributions coming from the theory of convex sets were investigated (e.g. uniform or gaussian distributions on compact convex sets). For more background on this, the reader is referred to the survey by Ábrego, Fernández-Merchant & Salazar [4] and to the book by Brass, Moser & Pach [11, Section 8]. We also point out that Sylvester’s question is actually related to several important conjectures in convex geometry [12, Chapter 3]. By a standard double-counting argument, one sees that , so the limiting density

 ck=limn→∞convk(n)/(nk)

is well defined and equal to the supremum of this ratio for . We apply the framework of flag algebras to order types and use the semi-definite method to obtain lower bounds on  for . As it turns out, the literature around the computation of  is vast: not only does  correspond to the last open case of a relaxation of Sylvester’s conjecture to all open sets of finite area, but as discovered by Scheinerman and Wilf [32], its value is determined by the asymptotic behaviour of the rectilinear crossing number of the complete graph, which has been extensively investigated. For this particular case, the best lower bound we could obtain is , which falls short of the currently best known lower bound, namely . This better lower bound is obtained by plugging results of Aichholzer et al. [6] and Ábrego et al. [1, 3] into an expression of the rectilinear crossing number found independently by Lovász et al. [27] and by Ábrego and Fernández-Merchant [2]. The best currently known upper bound on  is , and is due to Fabila-Monroy and López [20]. Nonetheless, our method allows us to strongly improve the known lower bounds on  and on .

###### Proposition 1.

We have and .

To the best of our knowledge, prior to this work the only known lower bounds on any constant  with  followed from a general and important result of Erdős and Szekeres [19], via a simple double counting argument. One indeed sees that  using that nine points in the plane must contain a convex pentagon (a result attributed to Makai by Erdős and Szekeres [19], the first published proof being by Kalbfleisch, Kalbfleisch & Stanton [23]). Similarly, as Szekeres and Peters [38] proved (using a computer-search) that the Erdős-Szekeres conjecture [19] is true for convex hexagons, one can use that seventeen points in the plane must contain a convex hexagon to infer that . The best upper bounds that we are aware of on these numbers are  and . We point out that Ábrego (personal communication) conjectured that . We again refer the interested reader to the survey and the book cited above [4, 11].

We prove Proposition 1 by a reformulation of limits of order types as positive homomorphisms from a so-called flag algebra of order types into  (see Proposition 9); this point of view allows a semidefinite programming formulation of the search for inequalities satisfied by limits of order types. Specifically, we argue that for any limit of order types

 ℓ(⋄5)⩾0.0608516andℓ(⋄6)⩾0.0018311,

where  is the order type of  points in convex position for any positive integer .

On a related topic, we can mention a recent application of flag algebras by Balogh, Lidický, and Salazar [9] to the (non rectilinear) crossing number of the complete graph. Their techniques differ from ours in that they use rotational systems instead of order types, and they use results about the crossing number of the complete graph for small numbers.

We now turn to another aspect of our work using flag algebras of order types. Probabilistic constructions often present extremal combinatorial properties that are beyond our imagination, a textbook example being the lower bound on Ramsey numbers for graphs devised by Erdős [16] in 1947. Sampling order types of a given size uniformly is of much interest to test conjectures and search for extremal examples (see e.g. [11, p. 326]). However the uniform distribution on order types seems out of reach as suggested by the lack of closed formulas for counting them, or heuristics to generate them, but we know of no formal justification of the hardness of approximation of this distribution. As it turns out, limits of order types can also be defined as families of probability distributions on order types with certain internal consistencies (see Proposition 7) and we exploit this interpretation to provide negative results on certain sampling strategies. Our second result obtained by the semidefinite method of flag algebras indeed shows that a broad class of random generation methods must exhibit some bias.

###### Proposition 2.

For any limit of order types  there exist two order types  and  of size  such that .

This unavoidable bias holds, in particular, for the random generation of order types by independent sampling of points from any finite Borel measure over , see creftype 8.

##### Representing limits by measures.

Given a finite Borel measure  over  and an order type , let  be the probability that  random points sampled independently from  have order type . It turns out (creftype 8) that is a limit of order types if and only if every line is negligible for . Going in the other direction, it is natural to wonder if geometric measures could serve as analytic representations of limits of order types, like graphons for limits of dense graphs. This raises several questions, here are some that we investigate: does every limit of order types enjoy such a realization? For those which do, what does the set of measures realizing them look like? In analogy with the study of realization spaces of order types [8, 14, 24, 33] we investigate how the weak topology on probability measures relates to the topology on limits.

We provide an explicit construction of a limit of order types that cannot be represented by any measure in the plane. For  let  be the probability distribution over  supported on two concentric circles with radii  and , respectively, where each of the two circles has -measure , distributed proportionally to the length on this circle. Because every line is negligible for , creftype 8 ensures that  is a limit of order types. We define  to be the limit of an arbitrary convergent sub-sequence of .

###### Proposition 3.

There exists no probability measure  such that .

The proof of Proposition 3 reveals that the sequence  is in fact convergent, so  is indeed an explicit example. Actually, an explicit description (for instance based on the reformulation of  used in the proof of Lemma 3.6) is possible, if tedious.

Let us now turn our attention to a family of limits that has a particularly nice realization space. Firstly, observe that nonsingular affine transforms preserve order types. Thus, if  is a measure that does not charge any line,  is a nonsingular affine transform, and  is the push-forward of  by , then . More generally, one can take  to be the restriction to  of an “adequate” projective map (we spell out the meaning of “adequate” in Section 3.1). Our main contribution in this direction is a projective rigidity result: we show that under natural measure theoretic conditions on , the realization space of  is essentially a point.

###### Theorem 4.

Let  and  be two compactly supported measures of  that charge no line and whose supports have non-empty interiors. If , then there exists a projective transformation  such that .

The converse of the statement of Theorem 4 is not true: it can fail for instance if the line mapped to infinity intersects the interior of . The approach we use to establish it actually provides a necessary and sufficient condition, expressed in terms of “spherical transforms” as defined in Section 3.1.

The third author conjectured [13] that the condition on  can be weakened.

###### Conjecture 5.

If  is a measure that charges no line and has support of Hausdorff dimension strictly greater than , then every measure that realizes  is projectively equivalent to .

It is easy to see that the conditions in creftype 5 are necessary. Recalling that  is the order type of  points in convex position, the limit of order types that for every  gives probability  to  can be realized by measures of Hausdorff dimension in . In particular, these are pairwise projectively inequivalent. More interestingly, we build a limit of order types  presenting projective flexibility, and which is interesting from the combinatorial geometry perspective: indeed,  is a nearly optimal lower bound for the aforementioned problem of Erdős, witnessing that .

###### Theorem 6.

There exists a limit of order types  that can be realized by a measure of Hausdorff dimension  for any  for which . Furthermore,  cannot be realized by a measure that is absolutely continuous with respect to the Lebesgue measure.

As in the rigidity result, the relation to the Hausdorff dimension seems fundamental: we observe that regular measures present a very different asymptotic behavior with respect to the density of .

## 2 Flag algebras

### 2.1 Reformulation of limits

Order types can be understood as equivalence classes of chirotopes under the action of permutations (see below). As such, they form an example of a model in the language of Razborov [30], and the theory of limits of order types is a special case of Razborov’s work. This section provides a self-contained presentation of the probabilistic and algebraic reformulations of limits of order types.

#### 2.1.1 Probabilistic characterization

Let  be a sequence of order types converging to a limit . Let , let  and let  be large enough so that for every . A simple conditioning argument yields that for any ,

 p(ω,ωn)=∑ω′∈Okp(ω,ω′)p(ω′,ωn). (1)

Indeed, the probability that a random sample realizes  is the same if we sample uniformly  points from a realization of , or if we first sample  points uniformly from that realization and next uniformly select a subset of  of these  points. It follows that any limit  of order types satisfies the following conditioning identities:

 ∀ω∈O,∀k⩾|ω|,ℓ(ω)=∑ω′∈Okp(ω,ω′)ℓ(ω′). (2)

As one notices, Equation 1 yields that the conditioning identities are equivalent to the (seemingly weaker) condition

 (3)

However, a stronger condition than Equation 2 is needed to characterize limits of order types, as we illustrate in Example 2.1. The split probability , where , and  are order types, is the probability that a random partition of a point set realizing  into two classes of sizes  and , chosen uniformly among all such partitions, produces two sets with respective order types  and . (In particular if .) We provide a detailed proof of the following proposition, but the reader familiar with the topic already sees the corresponding result for dense graph limits [26, Theorem 2.2].

###### Proposition 7.

A function  is a limit of order types if and only if

 ∀ω′,ω′′∈O,ℓ(ω′)ℓ(ω′′)=∑ω∈O|ω′|+|ω′′|p(ω′,ω′′;ω)ℓ(ω), (4)

and for every , the restriction  is a probability distribution on .

Before establishing Proposition 7, let us first point out that the product condition (4) implies the conditioning condition (2), as is seen by taking for  the (unique) order type of size .

###### Proof of Proposition 7.

We start by establishing the direct implication. The fact that is a probability distribution on  follows from the definition of a limit. As for Equation 4, fix two order types  and  in . Let  be a random order type sampled from  where . Let

 αn=p(ω′,ωn)p(ω′′,ωn)andβn=∑ω∈O|ω′|+|ω′′|p(ω′,ω′′;ω)p(ω,ωn).

Now, fix some point set  with order type . By the definition, the value  is the probability that  points sampled uniformly from  have order type . Now, on one hand,  equals the probability that two independent events both happens: (i) that a set  of  random points chosen uniformly from  has order type , and (ii) that another set  of  random points chosen uniformly from  has order type . On the other hand, observe that  equals the probability that (i) and (ii) happen and that  and  are disjoint. The difference  is therefore bounded from above by the probability that  and  intersect. Bounding from above the probability that  and  have an intersection of one or more elements by the expected size of  yields that

 ∣∣p(ω′,ωn)p(ω′′,ωn)−∑ω∈O|ω′|+|ω′′|p(ω′,ω′′;ω)p(ω,ωn)∣∣⩽E(|P′∩P′′|)=|ω′||ω′′||ωn|. (5)

Taking  in (5) we see that  satisfies Equation (4).

Conversely, suppose that  satisfies both conditions. For every integer , we pick a random order type  of size  according to . We assert that

 ∀ω∈O,P(limn→∞p(ω,rn)≠ℓ(ω))=0.

Since the number of order types is countable, we conclude that the random sequence  is convergent and has limit  with probability .

It remains to prove the assertion. Fix some  and assume that . Then

 E[p(ω,rn)]=∑ω′∈On2P(rn=ω′)p(ω,ω′)=∑ω′∈On2ℓ(ω′)p(ω,ω′)=ℓ(ω),

and

 Var(p(ω,rn))=E[p(ω,rn)2]−E[p(ω,rn)]2=∑ω′∈On2p(ω,ω′)2ℓ(ω′)−ℓ(ω)2.

By Equation 4,

 Var(p(ω,rn))=∑ω′∈On2p(ω,ω′)2ℓ(ω′)−∑ω′′∈O2|ω|p(ω,ω;ω′′)ℓ(ω′′).

Since , Equation 2 yields that . So we deduce that

 Var(p(ω,rn))=∑ω′∈On2⎛⎜⎝p(ω,ω′)2−∑ω′′∈O2|ω|p(ω,ω;ω′′)p(ω′′,ω′)⎞⎟⎠ℓ(ω′).

We observe that Inequality (5) obtained above is valid in general, and therefore

 Var(p(ω,rn))⩽∑ω′∈On2|ω|2n2ℓ(ω′)=|ω|2n2.

By Chebyshev’s inequality it thus follows that for any positive ,

 P(|p(ω,rn)−ℓ(ω)|>ε)⩽|ω|2ε2n2.

Therefore, for any and , the sum  is finite and hence the Borel-Cantelli lemma implies that with probability , only finitely many of the events  happen. Consequently, it holds with probability  that . ∎

Proposition 7 provides many examples of limits of order types.

###### Corollary 8.

Let  be a finite Borel measure over . The map  is a limit of order types if and only every line is negligible for .

###### Proof.

Assume that  is a limit of order types and let  be a sequence converging to . Let  be the order type of size . We have

 p(∴,μ)=ℓμ(∴)=limn→∞p(∴,ωn)=1,

so three random points chosen independently from  are aligned with probability , and consequently every line is negligible for .

Conversely, assume that  is a measure for which every line is negligible. For every integer  the restriction of  to  is a probability distribution. Moreover, for every order type  and every integer , we have

 ℓμ(ω)=∑ω′∈Omp(ω,ω′)ℓμ(ω′)

by sampling  points from  and sub-sampling  points among them uniformly. Proposition 7 then implies that  is a limit of order types. ∎

We conclude this section with a simple example of a function  that is not a limit of order types, and yet satisfies the conditioning identities and restricts on every  to a probability distribution.

###### Example 2.1.

Let  be the limit where convex order types have probability . Let be the limit defined before Proposition 3 on page 3. Set  to be .

First, note that if  and  are two limits of order types and  is any convex combination of  and , then the conditioning identities for  and  ensure that:

 ∀ω∈O,∀k>|ω|,ℓ3(ω)=∑ω′∈Okp(ω,ω′)ℓ3(ω′).

In particular, our function satisfies the conditioning identities. Similarly, it is immediate to check that for every , the restriction of to is a probability distribution on .

We show that  does not satisfy (4) by proving that

 ℓ(\vbox\includegraphics[page=5, height=19.9169% 29pt]{flag0.pdf})ℓ(∙)=132and∑|ω|=5p(\vbox% \includegraphics[page=5, height=19.916929pt]{flag0.pdf},∙ ;ω)ℓ(ω)=364.

To establish these equalities, we compute certain values of , which essentially boils down to computing two probabilities: (i) that the number of points on the outer circle equals that of the convex hull of the order type considered, and (ii) that the convex hull of these “outer points” contains the centre of the circle.

To compute (ii) requires to compute the probability that points chosen uniformly on a circle have the circle’s center in their convex hull. Let us condition on the lines formed by the points and the center of the circle (the lines being almost surely pairwise distinct). Each point is selected uniformly from the two possible, antipodal, positions on the corresponding line, and the initial choice of the lines does not matter. So the probability reformulates as: given pairs of antipodal points on the circle, how many of the -gons formed by one point from each pair contain the center? For the answer is  out of  configurations, while for the answer is  out of  configurations. We thus infer that

 ℓ⊙(\vbox\includegraphics[page=5, height% =19.916929pt]{flag0.pdf})=412414=116,ℓ⊙(\vbox% \includegraphics[page=3, height=19.916929pt]{flag0.pdf})=(52)12514=564,

and

 ℓ⊙(\vbox\includegraphics[page=2, height% =19.916929pt]{flag0.pdf})=512512=564.

It follows, on one hand, that

and on the other hand, that

 ∑|ω|=5p(\vbox% \includegraphics[page=5, height=19.916929pt]{flag0.pdf},∙ ;ω)ℓ(ω) =45ℓ(\vbox% \includegraphics[page=3, height=19.916929pt]{flag0.pdf})+25ℓ(\vbox\includegraphics[pag% e=2, height=19.916929pt]{flag0.pdf})+0ℓ(\vbox\includegraphics[page=1, height=19.916929pt]{flag% 0.pdf}) =25ℓ⊙(\vbox% \includegraphics[page=3, height=19.916929pt]{flag0.pdf})+15ℓ⊙(\vbox% \includegraphics[page=2, height=19.916929pt]{flag0.pdf}) =25×564+15×564 =364≠ℓ(\vbox% \includegraphics[page=5, height=19.916929pt]{flag0.pdf})ℓ(∙).

#### 2.1.2 Algebraic characterization

Recall that by (2) every limit  of order types satisfies

 ∀ω∈O,∀k⩾|ω|,ℓ(ω)=∑ω′∈Okp(ω,ω′)ℓ(ω′).

Now, let  be the set of all finite formal linear combinations of elements of  with real coefficients and consider the quotient vector space

 A=RO/KwhereK=vect{ω−∑ω′∈O|ω|+1p(ω,ω′)ω′:ω∈O}.

We define a product on  as follows:

 ∀ω1,ω2∈O,ω1×ω2=∑ω∈O|ω1|+|ω2|p(ω1,ω2;ω)ω, (6)

and we extend it linearly to . This extension is compatible with the quotient by  and therefore turns  into an algebra [30, Lemma 2.4].

An algebra homomorphism from  to  is positive if it maps every element of  to a non-negative real, and we define  to be the set of positive algebra homomorphisms from  to . Observe that any algebra homomorphism sends , the order-type of size one, to the real  as  is the neutral element for the product of order types.

###### Proposition 9 ([30, Theorem 3.3b]).

A map  is a limit of order types if and only if its linear extension is compatible with the quotient by  and defines a positive homomorphism from  to .

We equip  with a partial order , and write that with  if the image of  under every positive homomorphism is non-negative. The algebra  allows us to compute effectively with density relations that hold for every limit .

###### Example 2.2.

Let  be the order type on one point, and  the two order types of size four and , and  the three order types of size five, seen as elements of . It follows from the definitions and from Equation (2) that

 (7)

Since for any limit of order types  we have , the above implies that

Using again Equation (2), and the non-negativity of we obtain

and  for any limit of order types .

#### 2.1.3 Semi-definite method

Proposition 9 allows to search for inequalities over  by semidefinite programming. Let us give an intuition of how this works on an example. Here, we use the comprehensive list of all the order types of size up to , which was made available by Aichholzer  based on his work with Aurenhammer and Krasser [5] on the enumeration of order types. Throughout this paper, all non-trivial facts we use without reference on order types of small size can be traced back to that resource.

###### Example 2.3.

A simple (mechanical) examination of the  order types of size  reveals that for any . With Identity (2) this implies that or equivalently that . Observe that for any  and any (linear extension of a) limit of order types  we have by Proposition 9. We thus have at our command an infinite source of inequalities to consider to try and improve the above bounds. For instance, a tedious but elementary computation yields that

 (625\vbox\includegraphics[page=4, heigh% t=11.381102pt]{flag0.pdf} −11125\vbox% \includegraphics[page=5, height=11.381102pt]{flag0.pdf})2+2988191093750∑ω∈O8ω=∑ω∈O8aωω,

where  for every . This implies that for any limit of order types . The search for interesting combinations of such inequalities can be done by semidefinite programming.

### 2.2 Improving the semidefinite method via rooting and averaging

The effectiveness of the semidefinite method for limits of graphs was greatly enhanced by considering partially labelled graphs. We unfold here a similar machinery, using some blend of order types and chirotopes.

##### Partially labelled point sets, flags, σ-flags and Aσ.

A point set partially labelled by a finite set  (the labels) is a finite point set  together with some injective map . It is written  when we need to make explicit the set of labels and the label map. Two partially labelled point sets  and  have the same flag if there exists a bijection  that preserves both the orientation and the labelling, the latter meaning that for each . The relation of having the same flag is an equivalence relation, and a flag is an equivalence class for this relation. Again, we call any partially labelled point set a realization of its equivalence class, and the size  of a flag is the cardinality of any of its realizations.

A flag where all the points are labelled, i.e. where in some realization , is a -chirotope. (When a -chirotope coincides with the classical notion of chirotope.) Note that chirotopes correspond to types in the flag algebra terminology. Discarding the non-labelled part of a flag  with label set  yields some -chirotope  called the root of . A flag with root  is a -flag and by we mean the set of -flags. The unlabelling of a flag  with realization is the order type of .

Let  be a set of labels and  a -chirotope. We define densities and split probabilities for -flags like for order types. Namely, let , and  be -flags respectively realized by , and . The density of  in  is the probability that for a random subset  of size , chosen uniformly in , the partially labelled set  has flag . The split probability  is the probability that for a random subset  of size , chosen uniformly in , the partially labelled sets  and  have respective flags  and .

We can finally define an algebra of -flags as for order types. We endow the quotient vector space

 Aσ=RXσ/KσwhereKσ=vect{ω−∑ω′∈Xσ|ω|+1p(ω,ω′)ω′:ω∈Xσ}

with the linear extension of the product defined on  by .

###### Example 2.4.

Here are a few examples to illustrate the notions we just introduced. Letting  be the unique -chirotope with , there are exactly seven -flags on four points, three with a convex hull of size  and four with a convex hull of size . The densities of into each of these seven -flags with  points indicated below.

 p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=18, height=19.916929pt]{flag0.pdf}) =1, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=26, height=19.916929pt]{flag0.pdf}) =0, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=20, height=19.916929pt]{flag0.pdf}) =12, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=19, height=19.916929pt]{flag0.pdf}) =12, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=27, height=19.916929pt]{flag0.pdf}) =1, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=28, height=19.916929pt]{flag0.pdf}) =0, p(\vbox\includegraphics[page=17, he% ight=19.916929pt]{flag0.pdf},\vbox% \includegraphics[page=29, height=22.762205pt]{flag0.pdf}) =12.

Considering the quotient algebra, one sees that

 \vbox\includegraphics[page=21, height=22.762205pt]{flag0% .pdf} =\vbox\includegraphics[page=22, height=22% .762205pt]{flag0.pdf}+\vbox% \includegraphics[page=30, height=22.762205pt]{flag0.pdf} and \vbox\includegraphics[page=23, height=22.762205pt]{flag0% .pdf}
##### Rooted homomorphisms and averaging.

The interest of using the algebras  to study  relies on three tools which we now introduce. We first define an embedding of a -chirotope in an order type  to be a -flag with root  and unlabelling . We use random embeddings with the following distribution in mind: fix some point set realizing , consider the set  of injections such that  is a -flag, assume that , choose some injection  from  uniformly at random, and consider the flag of . We call this the labelling distribution on the embeddings of  in .

Next, we associate to any convergent sequence of order types  with limit , and for every -chirotope  such that , a probability distribution on . For every , the labelling distribution on embeddings of  in  defines a probability distribution  on mappings from  to ; specifically, for each embedding  of  in  we consider the map

 fθn:{Aσ→[0,1]τ↦p(τ,θn)

and assign to it the same probability, under , as the probability of  under the labelling distribution. As  is positive, the fact that converges as  for every  implies the weak convergence of the sequence  to a Borel probability measure  on  [30, Theorems 3.12 and 3.13]. Moreover, as is positive, the homomorphism induced by  determines the probability distribution  [30, Theorem 3.5].

We finally define, for every -chirotope , an averaging (or downward) operator  as the linear operator defined on the elements of  by , where  is the probability that a random embedding of  to  (for the labelling distribution) equals .

###### Example 2.5.

Here are a few examples of -flags, where  is the CCW chirotope of size :

For every -chirotope  and every limit of order types , we have the following important identity [30, Lemma 3.11]:

 (8)

which represents the fact that one can sample  by first picking a copy of  at random, and then, conditioning on the choice of , extend it to a copy of . Equation (8) in particular implies that for any  such that almost surely for , relatively to . It follows that for every limit of order types  and every -chirotope ,

 (9)

### 2.3 The semidefinite method for order types

The operator is linear, so for every , every , and every non-negative reals , we have

Every real (symmetric) positive semidefinite matrix  of size  can be written as where  are non-negative real numbers and  orthonormal vectors of . It follows that for every finite set of flags  and for every real (symmetric) positive semidefinite matrix  of size , we have , where  is the vector in  whose th coordinate equals the th element of  (for some given order). This recasts the search for a good “positive” quadratic combination as a semidefinite programming problem.

Let  be an integer, some target function, and  a finite list of chirotopes so that . For each , let be the -dimensional vector with th coordinate equal to the th element of . We look for a real  as large as possible subject to the constraint that there are  real (symmetric) positive semidefinite matrices , where  has size , so that

 (10)

The values of the real numbers  are determined by , the entries of the matrices , the splitting probabilities , where  and , and the probabilities , where . Moreover, finding the maximum value of  and the entries of the matrices  can be formulated as a semidefinite program.

##### Effective semidefinite programming for flags of order types.

In order to use a semidefinite programming software for finding a solution of programs in the form of (10), it is enough to generate the sets  and , the split probabilities , where  and , and the probabilities , where .

We generated the sets and the values by brute force up to . The only non-trivial algorithmic step is deciding whether two order types, represented by point sets, are equal. This can be done by computing some canonical ordering of the points that turn two point sets with the same order type into point sequences with the same chirotope. A solution taking time was proposed by Aloupis et al. [7]; the method that we implemented takes time  and seems to be folklore (we learned it from Pocchiola and Pilaud). For solving the semidefinite program itself, we used a library called CSDP [10]. The input data for CSDP was generated using the mathematical software SAGE [35].

##### Setting up the semidefinite programs.

In the rest of this section we work with  and use chirotopes labelled  with  being the empty chirotope, the only chirotope of size

two, and  the two chirotopes of size  depicted on the left, and  a fixed set of  chirotopes of size  so that . Note that since , what follows does not depend on the choices made in labelling . The vectors  described in the previous paragraph for this choice of  and ’s have respective lengths , , , , , , , , , , , , , , , , , , , , , , and .

##### Computations proving Proposition 1 and Proposition 2.

We solved two semidefinite programs with the above choice of parameters for  and  and obtained real symmetric positive semidefinite matrices  and  with rational entries so that

 ∑ω∈O8p(⋄5,ω)ω⩾∑i∈[24]\llbracketvTiMivi\rrbracketσi+15715211616602583691258254417031933722624∑ω∈O8ω,

and

 ∑ω∈O8p(⋄6,ω)ω⩾∑i∈[24]\llbracketvTiM′ivi\rrbracketσi+6755732468572598936893488147419103232∑ω∈O8ω.

The lower bounds on  and  then follow from Equation 2.

Assume (without loss of generality) that . Solving two semidefinite programs, we obtained real symmetric positive semidefinite matrices  and  as well as non-negative rational values  and  so that

 ∑j∈[20]dj⎛⎝ω6,j−132∑ω∈O8ω⎞⎠+∑i∈[24]\llbracketvTiMivi\rrbracketσi <0 and ∑j∈[20]d′j⎛⎝−ω6,j+118∑ω∈O8ω⎞⎠+∑i∈[24]\llbracketvTiM′ivi\rrbracketσi <0.

They respectively imply that there is no  such that, for every , or such that for every . Altogether this proves Proposition 2 with an imbalance bound of . The better bound of Proposition 2 is obtained by a refinement of this approach where the order types with minimum and maximum probability are prescribed; this requires solving over  semidefinite programs.

The numerical values of the entries of all the matrices  and coefficients  mentioned above can be downloaded from the web page http://honza.ucw.cz/proj/ordertypes/. In fact, the matrices  are not stored directly, but as an appropriate non-negative sum of squares, which makes the verification of positive semidefiniteness trivial. To make an independent verification of our computations easier, we created sage scripts called verify_prop*.sage, available from the same web page.

## 3 Representation of limits by measures

creftypecap 8 asserts that every probability (or finite Borel measure) over  that charges no line defines a limit of order types. Going in the other direction, we say that a measure  realizes a limit of order types  if . We examine here two questions: does every limit of order types enjoy such a realization and, for those that do, what does the set of measures realizing them look like? We answer the first question negatively in Section 3.2. We then give partial answers to the second question in Section 3.3 and Section 3.4. Every measure of  or on  that we consider is defined on the Borel -algebra.

### 3.1 Spherical geometry

If  is a measure that charges no line and  is an injective map that preserves orientations, then is another measure that realizes the same limit as . A map that preserves orientations must preserve alignments, and therefore coincides locally with a projective map. Note, however, that if we fix two points  and take a third point  “to infinity” in directions , the orientation of the triple  is different for  and for