On Levels in Arrangements of Surfaces in Three Dimensions
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Abstract
A favorite open problem in combinatorial geometry is to determine the worstcase complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n “pseudoplanes” or “pseudospherical patches” (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n ^{2.997}) vertices at any given level.
Keywords
kLevel Arrangements The kset problem1 Introduction
This question has relevance to the analysis of algorithms for a number of fundamental geometric problems [14, 20, 22, 24].Consider the number of vertices in the arrangement that have level equal to k. How large can this number be, asymptotically as a function of n and k?

The most famous case—dually related to the socalled kset problem —concerns lines in 2d. The early papers by Lovász [18] and Erdős et al. [16] showed that every arrangement of n lines has at most \(O(n\sqrt{k})\) vertices at level k, and furthermore, there exist arrangements of lines with Ω(nlogk) such vertices. Major improvements did not come for more than twenty years, until Dey [12] and Tóth [28] improved the upper bound to O(nk ^{1/3}) and the lower bound to \(n2^{\varOmega(\sqrt{\log k})}\).

For planes in 3d, Bárány, Füredi, and Lovász [5] were to first to obtain a nontrivial, subcubic upper bound of O(n ^{3−1/343})=O(n ^{2.9971}). After a series of improvements (in chronological order [1, 4, 13, 15]), Sharir, Smorodinsky, and Tardos [25] gave the current best upper bound of O(nk ^{3/2}). The current lower bound is \(nk2^{\varOmega(\sqrt{\log k})}\) [28].

For planes in 4d, Sharir [23] obtained the most recent upper bound O(n ^{4−1/18}) (improving a previous result by Matoušek et al. [21]). By a technique of Agarwal et al. [1], the bound can be made sensitive to k, namely, O(n ^{2} k ^{2−1/18}).

For hyperplanes in a fixed dimension d>3, Bárány et al.’s proof in combination with the multicolored Tverberg theorem [6, 29] yielded an upper bound of \(O(n^{d\alpha _{d}})\) for a very small α _{ d }=1/(2d)^{ d }. As Agarwal et al. [1] observed, the bound can be made sensitive to k, namely, \(O(n^{ \lfloor{d/2} \rfloor}k^{{ \lceil{d/2} \rceil}\alpha_{d}})\).

The more general case of nonlinear curves in 2d has also been studied extensively. Previous proofs for arrangements of lines, including Dey’s O(nk ^{1/3}) result, can be adapted to arrangements of pseudolines , where each pair of curves intersects at most once [26]. For more general families, however, new techniques are required. Tamaki and Tokuyama [27] were the first to suggest the approach of “cutting” curves and obtained an O(n ^{23/12}) upper bound on the number of vertices of any level in an arrangement of n pseudoparabolas , where each pair intersects at most twice. The bound for pseudoparabolas has eventually been reduced to O(nk ^{1/2}logk), after a series of improvements ([7, 2, 8, 19, 10] in chronological order). In 2005, the author [8] proposed a new, simple approach that yielded an \(\tilde{O}(nk^{11/2s})\) upper bound for general curve families, where each pair intersects at most a constant s number of times, and the \(\tilde{O}\) notation hides small, inverseAckermannlike factors; further improvements were also given for even values of s and for specific curve families (for example, graphs of fixeddegree polynomials in one variable). A followup paper [10] obtained yet more improvements.

For polyhedral surfaces comprising O(n) triangles in 3d, Agarwal et al. [1] gave an \(\tilde{O}(n^{2}k^{7/9})\) upper bound, which was improved to \(\tilde{O}(n^{2}k^{2/3})\) by Katoh and Tokuyama [17].
In this paper, we give upper bounds for nonlinear surfaces in 3d. The surface families that can be handled by our proof are fairly general, and include a certain definition of pseudoplanes and pseudotriangles , where the main condition is that each triple of surfaces intersects at most once, and pseudospherical patches , where the main condition is that each triple of surfaces intersects at most twice (see Sect. 2.3 for the precise general requirement and specific definitions used). For these particular surface families, our upper bound is O(n ^{3−1/286.97})=O(n ^{2.9966}), which can be made ksensitive, namely, O(nk ^{1.9966}) for pseudoplanes and \(\tilde{O}(n^{2}k^{0.9966})\) for the other families, by Agarwal et al.’s observations [1]. It should be emphasized that while this barely subcubic bound may not look very impressive, it is the first demonstration that a nontrivial result is possible, and with any luck, improvements might subsequently follow.^{1} (The reader is asked to take a historical perspective and compare our result with Bárány et al.’s for planes [5].) Rather than the specific bound, the proof technique itself should be regarded as the main contribution.
Why can’t previous techniques be adapted to handle general surfaces? The existing upperbound proofs for planes [4, 5, 13, 15, 25] and triangles [1, 17] in 3d are all similar in that they are based (in part) on Lovász’s original approach [18]. It is not unthinkable that this approach could work for pseudoplanes, although finding an appropriate extension of Lovász’s main lemma was posed by Agarwal et al. [1] some time ago and is still unanswered. In any case, an extension of this approach to more general surfaces in 3d (or even general curves in 2d) is less imaginable due to a variety of obstacles (for starters, Lovász’s lemma is typically applied to points in dual space, but no pointsurface duality is known). We therefore need a proof that is completely different from all previous proofs for planes. The author’s new technique for curves in 2d [8] turns out to be just what is needed, but the extension to 3d is not easy and requires a number of additional ideas, together with an intricate charging argument.
2 The Proof Plan
Note that a more common definition of pseudoplanes would omit condition (i). Later in Sect. 2.3, we will consider generalizations that would relax both conditions and allow for certain nonpseudoplane families.The surfaces are graphs of total bivariate functions such that (i) the intersection of each pair is an xmonotone curve, and (ii) the intersection of each triple is a single point.
2.1 The Previous Approach
As mentioned, the main approach is from the author’s previous paper [8]. The idea is to extend the problem by looking at the number of vertices at nearby levels. A nontrivial upper bound is then obtained by solving a recurrence/difference equation.
Given an arrangement of n pseudoplanes in ℝ^{3} and an integer k, let t _{ i } be the number of vertices in the arrangement with levels in the range (k−i,k+i). Let Δt _{ i }=t _{ i+1}−t _{ i } be the number of vertices with level k−i or k+i. Our problem is to bound t _{1}, the number of vertices in the arrangement at level k. (A more common version of the problem is to bound the combinatorial complexity of the klevel , defined as the boundary of the set of all points in ℝ^{3} with level at most k; the two versions are known to be asymptotically equivalent—e.g., see [3].)
We would like to derive a similar inequality in 3d (this time, with t _{ n }=O(n ^{3})). Unfortunately, there seems no obvious way to make the original charging argument work in 3d. Further ideas are needed ….
2.2 Reduction to 2d
As it turns out, the difficulties arising in 3d can be resolved by not working in 3d at all. More precisely, we consider the 2d subarrangement inside each of the n given surfaces, prove a general inequality in 2d, and then “sum up” to get the 3d inequality. (At least one previous proof, by Sharir et al. [25], was also in part based on summing up contributions from various 2d subproblems, though in a very different way; our idea here was indirectly inspired by Sharir et al.’s proof.)
The subarrangement within a given surface σ is a pseudoline arrangement when projected to the xyplane. In studying this subarrangement, we need to classify each pseudoline γ as one of two types, “red” or “blue”, depending on whether within σ, points above γ in the ydirection are above or below the surface defining γ in the zdirection.
We now need to prove an inequality in 2d that more generally applies to a bichromatic arrangement.
Definition 2.1
Consider an arrangement of pseudolines in ℝ^{2}, where each curve is colored red or blue. For a red curve γ, we say that a point p violates γ if p is strictly above γ; for a blue curve γ, p violates γ if p is strictly below γ. Define the level of p to be the number of curves violated by p. Call an intersection of two curves a monochromatic vertex if the two curves have the same color; call it a bichromatic vertex otherwise. Call a vertex at level in the range (k−i,k+i) an interior vertex, and a vertex at level k−i or k+i a boundary vertex.
Let \(t^{\mathrm{mo}}_{i}\) and \(t^{\mathrm{bi}}_{i}\) be the number of interior monochromatic and bichromatic vertices, respectively, and \(\Delta t^{\mathrm{mo}}_{i}=t^{\mathrm{mo}}_{i+1}t^{\mathrm{mo}}_{i}\) and \(\Delta t^{\mathrm{bi}}_{i}=t^{\mathrm{bi}}_{i+1}t^{\mathrm {bi}}_{i}\) be the number of boundary monochromatic and bichromatic vertices, respectively.
To overcome this problem, we need yet another idea: charge interior monochromatic vertices (\(t^{\mathrm{mo}}_{i}\)) not only to boundary vertices (\(\Delta t^{\mathrm{mo}}_{i}\) and \(\Delta t^{\mathrm{bi}}_{i}\)) but also to interior bichromatic vertices (\(t^{\mathrm{bi}}_{i}\)). Intuitively, if \(t^{\mathrm{mo}}_{i}\) is large, then \(t^{\mathrm{bi}}_{i}\) would be large too, as the example from Fig. 1 seems to suggest. We thus aim to prove the following.
Theorem 2.2
(Main Inequality)
In our 3d application, interior bichromatic vertices are fortunately rarer than interior monochromatic vertices, by a factor of 2. We claim that the above inequality implies a subcubic bound for a level in 3d:
Corollary 2.3
For any arrangement of n pseudoplanes in ℝ^{3}, the number of vertices at level k is \(O(n^{31/c_{0}})\), where \(c_{0}=\frac{3c_{1}}{2c_{2}}\), and c _{1} and c _{2} are the constants in the main inequality.
Proof
We apply Theorem 2.2 to the subarrangement inside each given surface, with the color scheme described above. Observe that the level of each vertex v in the 2d (bichromatic) subarrangement is equal to the level of v in the 3d arrangement. Each vertex v lies in three subarrangements. We claim that it is monochromatic in two of them and bichromatic in one of them: Consider the lower envelope of the three surfaces σ _{1},σ _{2},σ _{3} that define v. One edge of the envelope—say, σ _{1}∩σ _{2}—is to the left of v, and the other two are to the right of v, or vice versa. Then v is monochromatic in the subarrangement inside σ _{1} and the subarrangement inside σ _{2}, but bichromatic inside σ _{3}.
In Sect. 3, we will establish the 2d main inequality for a certain choice of constants c _{1} and c _{2}.
2.3 Generalization to Other Surface Families
 (∗)

The n given surfaces are graphs of total bivariate functions such that the 2d subarrangement within each surface (when projected to xyplane) forms a collection of O(n) xmonotone curve segments in a family with subquadratic cutting number.
Here, a family of curve segments has subquadratic cutting number if any N curve segments in the family can be cut into O(N ^{2−κ }) pseudosegments (where each pair intersects at most once), for some constant κ>0.
The cutting number in 2d arrangements was introduced by Tamaki and Tokuyama [27], who proved the first nontrivial results for pseudoparabolas (graphs of total univariate functions that pairwise intersect twice); the current best bound for pseudoparabolas has κ≈1/2 [2, 19]. Other curve families known to have subquadratic cutting number include pseudoparabolic segments (xmonotone curve segments that pairwise intersect at most twice) with κ=1/3 [7], and graphs of univariate degrees polynomial functions with κ≈1/2^{ s−1} [7, 19].

pseudotriangles , which we formally define as a collection of the graphs of n bivariate functions such that after nearvertical extensions are added, each 2d subarrangement forms a family of O(n) pseudosegments;

pseudospherical patches , which we formally define as a collection of the graphs of n bivariate functions such that after extensions are added, each 2d subarrangement forms a family of O(n) pseudoparabolic segments;

surfaces each having equation z=p(x)+ay for some degrees polynomial p(x) and constant a.
We can apply the same reduction to 2d (in the proof of Corollary 2.3, we should consider the lower envelope of σ _{1},σ _{2},σ _{3} only locally around v). All that is needed is a main inequality for 2d curve families with subquadratic cutting number. It is not obvious how to obtain such an inequality for general curves directly without blowing up the coefficient c _{2} beyond 2. One idea is to cut the curves first (which increases the number of endpoints) and then prove the same main inequality for pseudosegments (with n _{ i } redefined as the number of pseudosegment endpoints with level in [k−i,k+i]). The overhead term increases (to O(n ^{3−κ } i)), but this can still yield a subcubic result. However, for better results, we follow instead an idea from the previous paper [8] of not explicitly cutting the curves but charging features to certain “lenses”:
Definition 2.4
If two curves γ _{1} and γ _{2} intersect more than once, the part of γ _{1}∪γ _{2} between two consecutive intersection points is called a lens. We say that a lens is ilight if every vertical line segment inside the lens intersects at most i red curves and at most i blue curves.
We will prove the following generalization of the main inequality for arbitrary 2d xmonotone curves.
Theorem 2.5
(Main Inequality)
The same result would then hold for curve segments (with n _{ i } replaced by O(n)), since we can add nearvertical (upward if red, downward if blue) extensions at the endpoints to make the functions total—this increases Λ _{ i } by at most O(ni), as this generates at most O(i) extra lenses in Λ _{ i } incident to each endpoint.
Corollary 2.6
For any arrangement of n surfaces in ℝ^{3} satisfying (∗), the number of vertices at level k is \(O(n^{31/c_{0}})\) if κ>1/c _{0}, and O(n ^{3−κ }) if κ<1/c _{0}. Here, \(c_{0}=\frac{3c_{1}}{2c_{2}}\), where c _{1} and c _{2} are the constants in the main inequality.
3 Proof of the 2d Main Inequality
It remains to prove the (generalized) main inequality, Theorem 2.5. This is done in the next three subsections.
3.1 The Basic Charging Scheme
We start with a natural but more involved extension of the author’s charging argument for monochromatic 2d arrangements [8].
Let c be a positive constant, to be determined later. To avoid special cases, we treat the O(n _{ i }) points with level in [k−i,k+i] at x=±∞ as boundary vertices. We also pad the arrangement with (c+2)i+1 extra curves below and (c+2)i+1 extra curves above all the given curves of both colors, without creating any new intersections. We introduce some notation and terminology:
Definition 3.1

Call a vertex exceptional if it is a vertex of some lens in Λ _{ i }, and ordinary otherwise.

For any point v, let δ _{ v } be the number such that the level of v is k+δ _{ v }.

For a point v and a curve γ, let χ _{ v,γ } be +1 if v violates γ, and −1 otherwise.
 Given a vertex u defined by curves γ _{0} and γ _{1} and a vertex v defined by curves γ _{1} and γ _{2}, we say that v is forward w.r.t. u if γ _{2} lies between γ _{0} and γ _{1} at a vertical line slightly to the right (resp. left) of v, assuming that v is to the right (resp. left) of u. Otherwise, v is backward w.r.t. u. (Figure 2(left) shows an example of a forward vertex v and Fig. 2(right) shows an example of a backward vertex v.)

Given curves γ _{0} and γ _{1} of the same color and given a vertical line ℓ, let m(γ _{0},γ _{1},ℓ) be the number of curves of the same color as γ _{0},γ _{1} that lie between γ _{0} and γ _{1} at ℓ; let j(γ _{0},γ _{1},ℓ) be the number of curves of the opposite color as γ _{0},γ _{1} that lie between γ _{0} and γ _{1} at ℓ.

We say that (γ _{0},γ _{1},ℓ) is within range if j(γ _{0},γ _{1},ℓ)≤ci and both \(\delta_{\gamma_{0}\cap\ell},\delta_{\gamma_{1}\cap\ell}\in[i,+i]\). Otherwise, (γ _{0},γ _{1},ℓ) is out of range.
Observation 3.2
Proof
W.l.o.g., say γ _{0} is below v and is red (the other cases are similar). By the definition of levels in a bichromatic arrangement, \(m(\gamma_{0},\gamma_{1},\ell)j(\gamma_{0},\gamma_{1},\ell)\le\delta_{v} \delta_{\gamma_{0}\cap\ell}+1\). Since \(\delta_{v},\delta_{\gamma_{0}\cap\ell}\in[i,+i]\), we have m(γ _{0},γ _{1},ℓ)≤j(γ _{0},γ _{1},ℓ)+δ _{ v }+i+1≤(c+1)i+δ _{ v }+1≤(c+2)i+1. □
We now describe a scheme of charging interior monochromatic vertices to interior bichromatic vertices and boundary vertices (see Fig. 2).
Definition 3.3
 For an interior bichromatic vertex v, we say that u sends a charge to v if
 (A1)
(γ _{0},γ _{1},ℓ′) is within range for all vertical lines ℓ′ between u and v, and
 (A2)
v is forward w.r.t. u.
 (A1)

For a boundary (monochromatic or bichromatic) vertex v, we say that u sends ci−j(γ _{0},γ _{1},ℓ) charges to v if (A1) holds (regardless of whether (A2) holds).
Remarks
Lemma 3.4
 (i)
Each ordinary interior monochromatic vertex u sends at least 2ci charges.
 (ii)
Each interior bichromatic vertex v receives at most 4(c+1)i+O(1) charges.
 (iii)
Each boundary (monochromatic or bichromatic) vertex v receives at most 2c(c+2)i ^{2}+O(i) charges.
Proof
 (i)
Suppose u is defined by γ _{0} and γ _{1}. Imagine moving a vertical sweep line ℓ from left to right, starting at u. As ℓ passes through a bichromatic forward vertex on γ _{0}∪γ _{1} (w.r.t. u) while (γ _{0},γ _{1},ℓ) stays within range, the “counter” j(γ _{0},γ _{1},ℓ) increases by 1 and a charge is sent from u to that vertex. (On the other hand, as ℓ passes through a bichromatic backward vertex, the counter decreases by 1.) As soon as (γ _{0},γ _{1},ℓ) gets out of range, i.e., j(γ _{0},γ _{1},ℓ) reaches ci or ℓ passes through a boundary vertex on γ _{0}∪γ _{1}, terminate the sweep. In the latter case, ci−j(γ _{0},γ _{1},ℓ) charges are sent from u to the boundary vertex. Thus, at least ci charges are sent from u during this lefttoright sweep. Similarly, at least ci charges are sent during a righttoleft sweep.
 (ii)
Suppose v is defined by red curve γ _{1} and blue curve γ _{2}. By Observation 3.2, if v receives a charge from a vertex defined by γ _{1} and a red curve γ _{0} below v, then m(γ _{0},γ _{1},ℓ)≤(c+1)i+δ _{ v }+1. Thus, there are at most (c+1)i+δ _{ v }+1 candidates for γ _{0} below v, and by a symmetric argument at most (c+1)i−δ _{ v }+1 candidates for γ _{0} above v, yielding a total of at most 2(c+1)i+2 charges received by v from vertices on γ _{1}. Similarly, there are at most 2(c+1)i+2 charges from vertices on γ _{2}.
 (iii)Suppose v is defined by γ _{1} and γ _{2}. W.l.o.g., say γ _{1} is red. By Observation 3.2, j(γ _{0},γ _{1},ℓ)≥m(γ _{0},γ _{1},ℓ)−i−δ _{ v }−1 for red curves γ _{0} below v. Thus, v receives at most the following number of charges from vertices defined by γ _{1} and red curves below v: By a symmetric argument, the number of charges received by v from vertices defined by γ _{1} and curves above v is at most \((i\delta_{v})(ci) + \frac{c^{2}i^{2}}{2}+O(i)\), yielding a total of at most 2ci ^{2}+c ^{2} i ^{2}+O(i) charges received by v from vertices on‘γ _{1}. Similarly, there are at most 2ci ^{2}+c ^{2} i ^{2}+O(1) charges from vertices on γ _{2}.
3.2 Helpers
Definition 3.5
 For a bichromatic interior vertex v, we say that (v,γ _{0}) is a helper (at v) if
 (B1)
\(m(\gamma_{0},\gamma_{1},\ell)\le(c+1)i+\chi_{v,\gamma _{0}}\delta_{v}+1\), and
 (B2)
no intersection of γ _{0} and γ _{1} sends a charge to v.
 (B1)

For a (monochromatic or bichromatic) boundary vertex v, we say that (v,γ _{0}) forms ci−j(γ _{0},γ _{1},ℓ) helpers (at v) if the same conditions (B1) and (B2) hold.
By inspecting the proofs of Lemma 3.4(ii) and (iii), we immediately see that
Lemma 3.6
 (i)
Each interior bichromatic vertex v receives at most 4(c+1)i−[the number ofhelpers at v]+O(1) charges.
 (ii)
Each boundary (monochromatic or bichromatic) vertex v receives at most 2c(c+2)i ^{2}−[the number of helpers at v]+O(i) charges.
Is it always possible to find many helpers in an arrangement? A “canonical” example where there are no helpers at a vertex v is shown in Fig. 4(right), but in this example one can find helpers at nearby vertices. This suggests hope of a positive answer ….
We now classify helpers into a few specific types. The list of definitions below is somewhat elaborate, because of the desire to obtain better constants c _{1},c _{2} in our proof and to handle general nonpseudoline curves.
Definition 3.7
 For an interior bichromatic vertex v, we say that (v,γ _{0}) is a strong helper (from u) if
 (C1)
(γ _{0},γ _{1},ℓ′) is within range for all vertical lines ℓ′ between u and v, and
 (C2)
v is backward w.r.t. u.
 (C1)
 For an interior bichromatic vertex v, we say that (v,γ _{0}) is a moderate helper (from u) ifA moderate helper is further classified as a moderate forward or moderate backward helper depending on whether v is forward or backward w.r.t. u.
 (D1)
(γ _{0},γ _{1},ℓ) is within range, and
 (D2)
(γ _{0},γ _{1},ℓ′) is out of range for some vertical line ℓ′ between u and v, and
 (D3)
m(γ _{0},γ _{1},ℓ′),j(γ _{0},γ _{1},ℓ′)≤c _{3} i for all vertical lines ℓ′ between u and v.
 (D1)

For an interior bichromatic vertex v, any helper (v,γ _{0}) that is not strong or moderate are classified as a weak helper.

For a boundary vertex v, we say that (v,γ _{0}) forms ci−j(γ _{0},γ _{1},ℓ) moderate helpers (from u) if (D1)–(D3) hold. These moderate helpers are classified as moderate forward helpers (regardless of whether v is actually forward or backward w.r.t. u).
Remarks
Note that if (v,γ _{0}) is a strong or moderate helper from u, then γ _{0} cannot cross γ _{1} between u and v, because otherwise u would define a lens that is (c _{3} i)light (by condition (C1) or (D3)) and would thus be exceptional for a sufficiently large c _{3} (like in Fig. 3(left), with “(c+2)i+1” replaced by “c _{3} i”). For a similar reason, (v,γ _{0}) cannot be a strong or moderate helper from both a vertex u left of v and another vertex u′ right of v (like in Fig. 3(right)). Nor can we have both a strong or moderate helper (v,γ _{0}) from a vertex u left of v, and v receiving a charge from another vertex u′ right of v.
It can then be checked that each helper (v,γ _{0}) can indeed be only one of the classified types: strong, moderate forward, moderate backward, and weak; and furthermore, a strong or moderate helper is indeed a helper (because (B1) is implied by (C1) or (D1), and (B2) is implied by (C2) or (D2)).
The following lemma is useful in further improving constants. It turns out that strong helpers can not only reduce the number of charges received but also boost the number of charges sent. Furthermore, an abundance of moderate backward helpers automatically imply an abundance of moderate forward helpers.
Lemma 3.8
 (i)
Each ordinary interior monochromatic vertex u sends at least 2ci+[the numberof strong helpers from u] charges.
 (ii)
From each ordinary (not necessarily interior) monochromatic vertex u, the number of moderate forward helpers is at least the number of moderate backward helpers.
Proof
 (i)
This follows by inspecting the proof of Lemma 3.4(i). During the sweep for vertex u, when ℓ passes through a bichromatic backward vertex v on γ _{0}∪γ _{1}, we get a strong helper at v from u and the counter j(γ _{0},γ _{1},ℓ) decreases by 1, allowing for one more subsequent increment (and thus a charge to one more interior vertex) or a lower final value for j(γ _{0},γ _{1},ℓ) (and thus an extra charge to a boundary vertex).
 (ii)
Perform the same lefttoright sweep starting at u, except this time the sweep is terminated only when j(γ _{0},γ _{1},ℓ) or m(γ _{0},γ _{1},ℓ) exceeds c _{3} i. The regions swept by ℓ during which (γ _{0},γ _{1},ℓ) is within range form a union of disjoint “windows”. Take one such window, excluding the initial window containing u (there is no initial window if u is not an interior vertex). When ℓ passes through a bichromatic forward (resp. backward) vertex v on γ _{0}∪γ _{1} within this window, we get a moderate forward (resp. backward) helper at v from u and the counter j(γ _{0},γ _{1},ℓ) increases (resp. decreases) by 1. At the right boundary of the window, j(γ _{0},γ _{1},ℓ) reaches ci or ℓ passes through a boundary vertex v on γ _{0}∪γ _{1}. In the latter case, we get ci−j(γ _{0},γ _{1},ℓ) moderate forward helper at v. Thus, there are at least as many moderate forward helpers as moderate backward helpers within each window. A similar argument holds for a righttoleft sweep.
Let H _{strong}, H _{for}, H _{back}, and H _{weak} denote the number of strong, moderate forward, moderate backward, and weak helpers, respectively.
3.3 A More Sophisticated Charging Scheme
We now prove the abundance of helpers by devising a second charging scheme, this time, with charges sent from interior bichromatic vertices to helpers. Let α>2/c and β∈(α,1−α) be constants, to be set later.
Sweeping
Take an ordinary interior bichromatic vertex v, defined by red curve γ _{1} and blue curve γ _{2}. W.l.o.g., say γ _{1} is below γ _{2} slightly to the left of v (the other case is symmetric). Let h be the number of helpers of the form (v,γ) with γ red and below v, or blue and above v.

Let j _{1} (resp. j _{2}) be the number of blue (resp. red) curves between γ _{1} and γ _{2}.

Let m _{ σ } (resp. \(m'_{\sigma}\)) be the number of bichromatic forward (resp. backward) vertices on γ _{ σ } between ℓ and v (σ∈{1,2}). Let m=m _{1}+m _{2}.

Let p _{ σ } (resp. \(p_{\sigma}'\)) be the number of monochromatic forward (resp. backward) vertices on γ _{ σ } between ℓ and v.
Observation 3.9
\(j_{1}+m_{1}',\,j_{2}+m_{2}'\le m+i+\delta_{v}+1 < ci\).
Proof
By the definition of levels, \(m_{2}+p_{2}'m_{2}'p_{2}\ge\delta_{\gamma_{2}\cap\ell}\delta_{v}1\). Thus, \(j_{1}=m_{1}+p_{2}m_{1}'p_{2}'\le m_{1}+m_{2}m_{1}'m_{2}'+\delta_{v}\delta_{\gamma _{2}\cap \ell}+1\). Since \(\delta_{v},\delta_{\gamma_{2}\cap\ell}\in[i,+i]\), we have \(j_{1}+m_{1}'\le m+i+\delta_{v}+1\le(1\alpha)ci + 2i < ci\). The other inequality for \(j_{2}+m_{2}'\) is similar. □
Remark
Note that during the righttoleft sweep, γ _{1} cannot cross γ _{2}, because otherwise v would define a lens that is (ci)light (by Observation 3.9) and would thus be exceptional.
Actually, in the above sweep, ℓ cannot hit a boundary vertex on γ _{1}∪γ _{2}. To see this, suppose ℓ reaches a boundary vertex w on γ _{1} (the other case is similar). For at least (c+1)i+δ _{ v }+1−h red curves γ below v, some intersection y _{ γ } of γ and γ _{1} sends a charge to v (by the definition of helpers and the number h); and y _{ γ } must be to the right of w (in order for y _{ γ } to send a charge to v). Now, γ must cross the vertical line segment between w and γ _{2}∩ℓ, or cross γ _{2} between γ _{2}∩ℓ and v, because otherwise γ would cross γ _{1} between w and y _{ γ }, and y _{ γ } would define a lens that is (ci)light (since j _{1} and j _{2} always stay below ci by Observation 3.9) and would thus be exceptional. It follows that \(j_{2}+m_{2}'\) reaches at least (c+1)i+δ _{ v }+1−h, and by Observation 3.9, m reaches at least ci−h>(1−α)ci−h−1: a contradiction.
Charging
For each of the h helpers of the form (v,γ) with γ red and below v, or blue and above v, as initialization, v sends 1 unit of charge to the helper.
Lemma 3.10

Case 0: w is an exceptional vertex.

Case 1: (w,γ _{2}) is a strong or moderate backward helper.

Case 2: (v,γ _{0}) is a strong, moderate backward, or weak helper.

Case 3: At some vertical line ℓ′ between w and v, we have γ _{0} below γ _{1}, and there are more than ci blue curves or more than (c+2)i+1 red curves between γ _{0} and γ _{1}.
Proof
Suppose that Case 3 does not hold. Then at every vertical line ℓ′ between w and v, if γ _{0} is below γ _{1}, there are at most ci blue curves and at most (c+2)i+1 red curves between γ _{0} and γ _{1}. In particular, if γ _{0} is below γ _{1} at ℓ, there are at most ci≤(c+1)i+δ _{ v }+1 blue curves between γ _{0} and γ _{1} at ℓ.
We may assume that γ _{0} and γ _{1} do not cross between w and v (and thus γ _{0} is indeed below γ _{1} at ℓ), because otherwise w would define a lens that is ((c+2)i+1)light and we would be in Case 0.
If v receives a charge from a vertex u on γ _{0}, or if (v,γ _{0}) is a moderate forward helper from u, then (w,γ _{2}) is a strong or moderate backward helper from u and we are in Case 1 ((D1) and (D3) hold since j _{1} and j _{2} stay below ci by Observation 3.9, as one can see from Fig. 5(right) for a sufficiently large c _{3}>2c+2). On the other hand, if (v,γ _{0}) is a strong, moderate backward, or weak helper, then we are in Case 2. □

In Case 0, v sends 1 unit of charge to the vertex w itself.

In Case 1, v sends 1 unit of charge to the helper (w,γ _{2}).

In Case 2, v sends 1 unit of charge to the helper (v,γ _{0}).

In Case 3, note that for at least (c+1)i+δ _{ v }+1−h red curves γ below v, some intersection y _{ γ } of γ and γ _{1} sends a charge to v (by definition of helpers and the number h). If y _{ γ } is to the right of w, then γ must cross the vertical line segment between w and γ _{2}∩ℓ, or cross γ _{2} between γ _{2}∩ℓ and v, because otherwise γ would cross γ _{1} between w and y _{ γ }, and y _{ γ } would define a lens that is (ci)light (since j _{1} and j _{2} always stay below ci by Observation 3.9) and would thus be exceptional. Thus, by excluding at most \(j_{2}+m_{2}'\) candidates for γ, we can ensure that y _{ γ } is to the left of w.
Since y _{ γ } sends a charge to v, there are at most ci blue curves and at most (c+2)i+1 red curves between γ and γ _{1} at ℓ′ (by (A1), with Observation 3.2). In particular, γ must be above γ _{0} at ℓ′, due to the condition stated in Case 3. Thus, some intersection z _{ γ } of γ and γ _{0} must be backward w.r.t. y _{ γ }, and so (z _{ γ },γ _{1}) is a strong helper from y _{ γ } (see Fig. 6(right); condition (C1) holds for this helper because y _{ γ } sends a charge to v). The number of candidates for γ is at least \((c+1)i+\delta_{v}+1hj_{2}m_{2}'\ge cihm\) by Observation 3.9. We make v send \(\frac{1}{\beta ci}\) units of charge to each such strong helper.
This completes the description of the righttoleft sweeping and charging process for v. We perform a similar process for v, this time, sweeping from left to right.
Analysis

Each exceptional vertex w receives O(i) charges due to Case 0, since m _{1},m _{2}≤(1−α)ci implies that given w, there are O(i) candidates for γ _{2}, and thus for v.

Each strong helper receives 1 charge initially, at most 1 charge due to Case 1, at most 1 charge due to Case 2, and at most \((1\alpha)ci\frac{1}{\beta ci} = (1\alpha)/\beta\) charges due to Case 3, since m _{1},m _{2}≤(1−α)ci implies that given (z _{ γ },γ _{1}), there are at most (1−α)ci candidates for γ _{2}, and thus for v.

Each moderate forward helper receives 1 charge during initialization and no more afterwards.

Each moderate backward helper receives 1 charge during initialization, at most 1 charge due to Case 1, and at most 1 charge due to Case 2.

Each weak helper receives 1 charge during initialization, and at most 1 charge due to Case 2.
The coefficient of the \(t^{\mathrm{bi}}_{i}\) term here is strictly less than 2 for a sufficiently large choice of the parameter c. The proof of Theorem 2.5 is now complete. For Corollaries 2.3 and 2.6, we have \(c_{0}=\frac{3(c+2)}{0.3101022/c}\), which is below 286.97 by setting c=13.8312, with c _{1}=15.8312 and c _{2}<1.834499. (Note that indeed α>2/c.)
4 Final Remarks
Just like in earlier work on levels in arrangements of curves, we have shown how a nontrivial cutting number bound translates to a nontrivial level bound for any family of surfaces in 3d. In light of this result, we reiterate the following open problem [7]: do general fixeddegree algebraic curves in the plane have a subquadratic cutting number? If so, a subcubic level bound would immediately follow for graphs of fixeddegree bivariate polynomial functions in 3d.
Perhaps it might be possible to obtain slight improvements on the constants in the 2d main inequality by lengthening the proof with an even more detailed case analysis, but it would be more desirable to find a simpler yet smarter charging argument that could yield more drastic improvements. An intriguing question is to determine what is the smallest value c _{2} attainable in the 2d inequality. Alternatively, can one prove the 3d inequality directly?
It is doubtful that our approach could improve known upper bounds for levels for planes (i.e., the kset problem) in 3d. A more intriguing direction to pursue would be the case of hyperplanes in higher dimensions, where the previous upper bounds are very weak. Unfortunately, our reduction to 2d fails to yield o(n ^{ d }) bounds as soon as the dimension d reaches 4, because there could be as many bichromatic vertices (\(t^{\mathrm {bi}}_{i}\)) as monochromatic vertices (\(t^{\mathrm{mo}}_{i}\)) on average in the resulting 2d subarrangements, and so we need the coefficient c _{2} to be strictly less than 1—an impossible demand (as the example in Fig. 1 indicates). Still, it is possible to adapt the approach of this paper to obtain new ksensitive upper bounds for hyperplanes in 4d for a certain range of k values, as the author has shown in a recent paper [11]. New ksensitive upper bounds might also be possible in higher dimensions if we could somehow get c _{2} closer to 1. In 2d and 3d, our approach has been shown [11] to lead to new results for a bichromatic version of the kset problem.
Footnotes
Notes
Acknowledgement
This work was supported in part by an NSERC grant.
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