Limits of Order Types††thanks: This work was partially supported by ANR blanc PRESAGE (ANR–11-BS02–003).
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 .
Limits of structures, flag algebra, geometric measure theory, Erdős-Szekeres theorem, Sylvester’s problem.
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 , 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 .
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 . 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.
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
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  in 1973 (see also ): “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  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
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 , 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.  and Ábrego et al. [1, 3] into an expression of the rectilinear crossing number found independently by Lovász et al.  and by Ábrego and Fernández-Merchant . The best currently known upper bound on is , and is due to Fabila-Monroy and López . Nonetheless, our method allows us to strongly improve the known lower bounds on and on .
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 , 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 , the first published proof being by Kalbfleisch, Kalbfleisch & Stanton ). Similarly, as Szekeres and Peters  proved (using a computer-search) that the Erdős-Szekeres conjecture  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
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  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  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.
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 .
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.
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  that the condition on can be weakened.
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 .
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 , 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 ,
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:
As one notices, Equation 1 yields that the conditioning identities are equivalent to the (seemingly weaker) condition
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].
A function is a limit of order types if and only if
and for every , the restriction is a probability distribution on .
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
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
Conversely, suppose that satisfies both conditions. For every integer , we pick a random order type of size according to . We assert that
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
By Equation 4,
Since , Equation 2 yields that . So we deduce that
We observe that Inequality (5) obtained above is valid in general, and therefore
By Chebyshev’s inequality it thus follows that for any positive ,
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.
Let be a finite Borel measure over . The map is a limit of order types if and only every line is negligible for .
Assume that is a limit of order types and let be a sequence converging to . Let be the order type of size . We have
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
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.
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:
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
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
It follows, on one hand, that
and on the other hand, that
2.1.2 Algebraic characterization
Recall that by (2) every limit of order types satisfies
Now, let be the set of all finite formal linear combinations of elements of with real coefficients and consider the quotient vector space
We define a product on as follows:
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 .
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
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 111http://www.ist.tugraz.at/aichholzer/research/rp/triangulations/ordertypes/ based on his work with Aurenhammer and Krasser  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.
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
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, -flags and .
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
with the linear extension of the product defined on by .
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.
Considering the quotient algebra, one sees that
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
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 .
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]:
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 ,
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
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. ; 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 . The input data for CSDP was generated using the mathematical software SAGE .
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
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
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