New estimates for convex layer numbers
NNEW ESTIMATES FOR CONVEX LAYER NUMBERS
GERGELY AMBRUS, PETER NIELSEN, AND CALEDONIA WILSON
Abstract.
We study evenly distributed families of point sets containedin B d , the d -dimensional unit ball. We show that for such families, L ( X ) ≥ Ω( | X | /d ) holds, with the bound being sharp. On the otherhand, building on earlier results, we prove that for such families of sets, L ( X ) ≤ O ( | X | /d ) holds for d ≥ d ≥ X ⊂ B d with L ( X ) = Θ( | X | /d − / ( d d − ) ), showing thatthe upper bound is nearly tight. Introduction
Let X be a finite point set in R d . The convex hull of X , denoted byconv X , is a convex polytope whose vertices are the extreme points of X .The set of extreme points will be denoted by V ( X ). We call V ( X ) the firstconvex layer of X .Consider the following peeling process. Start with X . In each step, deletethe extreme points of the current set from it – these constitute the convexlayers of the set X . In finitely many steps, we reach the empty set. Thenumber of steps is L ( X ), the layer number of X . We note that in somesources, the layer number is referred to as the convex depth of X . Moreover,for x ∈ X , the depth of x is the number of the step of the peeling process inwhich x is deleted.Layer numbers were first studied in 1985 by Chazelle [2] from the algorith-mic point of view, who gave an optimal, O ( n log n ) running time algorithmfor computing the convex layers of an n -element planar point set.Almost 20 years later, Dalal [3] studied the layer number of random pointssets. He proved that if X is a set of n random points chosen independentlyfrom the d -dimensional unit ball, then L ( X ) = Θ( n / ( d +1) ). We note thatstudying the convex hull of random point sets (i.e. the first layer) datesback much earlier, see the works of R´enyi and Sulanke [8] and Raynaud [7].Har-Peled and Lidick´y [5] showed that the layer number of the planar √ n × √ n grid is Θ( n / ), with the analogous question for higher dimensionsstill being open. The conjectured asymptotic bound for the general case isΘ( n / ( d +1) ), the same as for random point sets.In his Master’s Thesis [6], W. Joo studied the layer numbers of α -evenlydistributed point sets in R d . For such sets, he proved the upper bound Date : June 9, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Layer numbers, peeling process, convex hull, extreme points,geometric processes.Research of the first author was supported by NKFIH grant PD125502. Work of thethird author was supported by the Hungarian – American Fulbright Commission. a r X i v : . [ m a t h . M G ] J un GERGELY AMBRUS, PETER NIELSEN, AND CALEDONIA WILSON L ( X ) ≤ O | X | d +12 d . Moreover, he showed that this bound is sharp for d = 2.These results are going to be published in the article [4] written jointlywith I. Choi and M. Kim, which serves as the starting point for our currentresearch.In this paper, we are going to study evenly distributed families of pointsets contained in B d , the d -dimensional unit ball. We prove that for suchfamilies of sets in R d with d ≥ L ( X ) ≥ Ω( | X | d ), which is a sharp bound,and L ( X ) ≤ O ( | X | d ) (which improves the current upper bound for d ≥ d ≥ X ⊂ B d with L ( X ) = Θ( | X | (1 − / d )2 /d ), showing that the upper bound is nearly tight.Interestingly, the existence of an evenly distributed family of point sets X in R satisfying L ( X ) = Θ( | X | ) would also lead to a construction with L ( X ) = Θ( | X | d ) for all d ≥ d -dimensional Euclidean space R d with the origin o . By distance, werefer to the Euclidean distance between two points, denoted by | . | . As usual, B d denotes the unit ball in R d , with S d − , the unit sphere in R d , being itsboundary. As is well known,(1) s d − = dκ d , where s d − denotes the surface area of S d − , and κ d gives the volume of B d (see [1]). For basic definitions regarding convexity we refer to [9].The article [4] estimates layer numbers of α -evenly distributed sets, whichare defined as follows. Definition 1.1 ([4]) . Let X be a finite point set in a unit ball in R d . For aconstant α > , we say X is α -evenly distributed if for every Euclidean ball D with positive volume, | X ∩ D | ≤ (cid:6) α | X | Vol ( D ) (cid:7) For a set A ⊂ R d , let µ ( A ) denote the minimum (Euclidean) distancebetween different points of A (thus, µ ( A ) = 0 iff A has points of multiplicitylarger than 1). The following assertion shows that being α -evenly distributedis implied by a minimum distance condition. Lemma 1.1 (Lemma 2.3, [4]) . For every d ≥ , there exists a continuousbijection f d : R > → R > such that if X ⊂ B d is a finite point set satis-fying µ ( X ) ≥ βn − /d with some constant β > , then X is f d ( β ) -evenlydistributed. In fact, being α -evenly distributed is equivalent to the above property. Inthe current work, we study point sets satisfying this latter condition: theirminimum distance is asymptotically as large as possible. Definition 1.2.
A family of sets X , X , X , · · · ⊂ B d is said to be evenlydistributed if | X i | → ∞ and µ ( X i ) = Θ( | X i | − /d ) . The implied constant in Θ( | X i | − /d ) depends on d .By Lemma 1.1, every evenly distributed family is α -evenly distributed forsome parameter α. In fact, the two definitions are equivalent to each other.
EW ESTIMATES FOR CONVEX LAYER NUMBERS 3
A key property that we use repeatedly is that if X ⊂ B d is a memberof an evenly distributed family in R d with | X | = n , then any ball of radius O ( n − /d ) contains at most O (1) points of X . This follows by a standardvolume estimate. 2. Well-separated sets
The quintessential example of evenly distributed families is given by thefollowing standard definition.
Definition 2.1.
A point set X ⊂ R d is δ -separated if µ ( X ) ≥ δ . For A ⊂ R d and X ⊂ A , we call X to be maximal δ -separated in A if X is δ -separated, and for any point y ∈ A , | x − y | < δ holds for some x ∈ X . In other words, no further point of A may be added to X without loosingthe δ -separated property. It is often used that maximal δ -separated sets arealso δ -nets: Definition 2.2.
Let A ⊂ R d and X ⊂ A . Then X is a δ -net in A if forevery point y ∈ A , there exists x ∈ X such that | x − y | ≤ δ . We are going to make extensive use of the above notions with A beingthe unit sphere S d − . In this case, the following well-known bound holds forthe cardinality of maximal δ -separated sets. Lemma 2.1.
Let X be a maximal δ -separated set in S d − . Then X is a δ -net, and | X | = Θ( δ − ( d − ) . For self-containedness, we include the following sketch of the proof.
Proof.
Since X is a maximal δ -separated set in S d − , for any point y ∈ S d − , | x − y | < δ holds for some x ∈ X . This shows that X is indeed a δ -net in S d − .Additionally, since X is δ -separated set, Euclidean balls of radius δ/ X are pairwise non-overlapping. For sufficientlysmall values of δ , the intersection of such a ball with S d − is well approx-imated by a spherical cap of radius δ/
2, which has surface area Θ( δ d − ).Since these spherical caps are pairwise non-overlapping, we readily obtainthat | X | = O ( δ − ( d − ). On the other hand, since X is a δ -net in S d − , ballsof radius δ centred at the points of X cover S d − , and the same estimatesshow that | X | = Ω( δ − ( d − ). (cid:3) Clearly, the above asymptotic estimate remains true for any sphere witha fixed radius.Without additional assumptions, the cardinality of a δ -separated set inthe unit ball B d may be as large as Θ( δ − d ). However, if the set is also inconvex position, we may give a stronger estimate. Lemma 2.2.
For any point set X ⊂ B d which is δ -separated and in convexposition, | X | ≤ O ( δ − ( d − ) . GERGELY AMBRUS, PETER NIELSEN, AND CALEDONIA WILSON
Proof.
We show that the point set may be moved outwards to S d − suchthat pairwise distances between points do not decrease, and then use thebound on the cardinality of a set in S d − provided by Lemma 2.1.Assume x ∈ X is in the interior of B d , that is, | x | <
1. Let P = conv X .Since x is a boundary point of P , there exists an outer normal direction u at x (see e.g. [9]). That is, u ⊥ + x is a supporting hyperplane of P , with u pointing away from P . Then for any positive λ > y ∈ X , y (cid:54) = x ,we have that | y − x | < | y − ( x + λu ) | . Set λ (cid:48) so that | x + λ (cid:48) u | = 1, andlet x (cid:48) = x + λ (cid:48) u . Then if X (cid:48) is obtained from X by replacing x by x (cid:48) , allpairwise distances of X (cid:48) are at least as large as the corresponding distancein X . By repeating the same process for each point of X in the interiorof B d , we obtain a δ -separated point set in S d − . The distance conditionimplies that distinct points remain distinct. By Lemma 2.1, the cardinalityof this point set cannot exceed Θ( δ − ( d − ). (cid:3) The sharp lower bound on the layer number
Our first results establish the sharp lower bound for evenly distributedfamilies in B d . Theorem 3.1.
Assume that { X i } ∞ i =1 ⊂ B d is an evenly distributed familyin R d . Then L ( X i ) ≥ Ω( | X i | /d ) .Proof. By Definition 1.2, µ ( X i ) = Θ( | X i | − /d ). Thus, Lemma 2.2 impliesthat each convex layer of X i may have at most O ( | X | d − d ) vertices. Hence,we obtain that L ( X ) = Ω( | X | /d ). (cid:3) It is not hard to prove that the above lower bound may be achieved.
Proposition 3.2.
For every d ≥ , there exists an evenly distributed family ( X i ) ∞ with L ( X i ) = Θ( | X i | /d ) .Proof. Let S (1 /
2) denote the sphere of radius 1/2 centered around the originin R d . For every i ≥
1, let δ i = 1 /i , and let D i be a maximal δ i -separated seton S (1 / | D i | = Θ( δ − ( d − i ) = Θ( i ( d − ). Clearly, D i is in convex position. To construct X i , we let X i = i − (cid:91) j =0 (cid:18) ji (cid:19) D i . Then, | X i | = i | D i | = Θ( i d ), and thus δ i = Θ( | X i | − /d ). The shell structureof the construction implies that L ( X i ) = i . We only have to show that µ ( X i ) = Θ( | X i | − /d ) = Θ( i − ) holds. The distance between points in thesame layer of X is, by definition, at least δ i . Finally, the distance betweenpoints on different layers is at least the difference between the radii of theselayers, which is not less than 1 /i = δ i . Therefore, X i is δ i -separated, andthe assertion follows. (cid:3) An upper bound on the layer number
It is easy to see that for d = 1, the layer number of any set X of distinctpoints in R is Θ( | X | ). In the planar case, Choi, Joo, and Kim [4] proved EW ESTIMATES FOR CONVEX LAYER NUMBERS 5 that if X is an α -evenly distributed set with a parameter α >
1, then L ( X ) = O ( | X | / ), and this bound may be achieved. They also extended thebound to higher dimensions by showing that if X is an α -evenly distributedset in R d , then L ( X ) = O ( | X | ( d +1) / d ), and they conjectured that this boundis in fact sharp. We improve their estimate to the following asymptotic upperbound: Theorem 4.1.
Assume that { X i } ∞ i =1 ⊂ B d is an evenly distributed familyin R d . Then L ( X i ) ≤ O ( | X i | /d ) . The proof follows that in [4]. The key tool is the following statementtherein.
Lemma 4.2 ([4]) . If K and K are two convex bodies in R d such that K ⊆ int( K ) and X is a finite point set in K , then L ( X ) ≤ max {| X ∩ C | : C is a cap of K \ K } + L ( X ∩ K ) . Here a cap of K \ K is the intersection K ∩ H + , where H is a supportinghyperplane of K defining the corresponding closed halfspaces H − and H + ,of which H − contains K .The proof of Lemma 4.2 may be found in [4]. The key idea is that if a cap C satisfies C ∩ X (cid:54) = ∅ , then C must also contain a point of V ( X ). Thus, if m denotes the maximal number of points of X contained in a cap of K \ K ,then in m steps of the peeling process, no point of X may remain in anycap of K \ K . Proof of Theorem 4.1.
Let X be a member of an evenly distributed familyin B d with | X | = n . Set N = (cid:106) n /d (cid:107) , and for j ∈ { , . . . , N } , let B j =(1 − jn /d ) B d . For each 0 ≤ j ≤ N −
1, let C j be a cap of B j \ B j +1 whichcontains the maximum number of points of X among all caps of B j \ B j +1 .An elementary calculation reveals that the radius of the base of C j is atmost (cid:113) − (1 − n /d ) = O ( n − /d )while its height is n − /d = o ( n − /d ). Thus, C j is contained in a ball of radius O ( n − /d ). According to the remark following Definition 1.2, | C j ∩ X | ≤ O (1).Furthermore, since the radius of B N is at most n − /d < n − /d , we also have | B N ∩ X | ≤ O (1). Therefore, by Lemma 4.2, L ( X ) ≤ | X ∩ B N | + N − (cid:88) j =0 | X ∩ C j |≤ O (1) + N − (cid:88) j =0 O (1) ≤ O ( n d ) . (cid:3) Tangent polytopes
The following notion plays a key role in the subsequent construction ofpoint sets with large layer number.
GERGELY AMBRUS, PETER NIELSEN, AND CALEDONIA WILSON
Definition 5.1.
Let X ⊂ S d − be a finite point set spanning R d with ∈ conv X . Define the tangent polytope P ( X ) of X as P ( X ) = ∩ x ∈ X H − ( x ) where for each x ∈ X , H ( x ) is the supporting hyperplane of S d − at x , and H − ( x ) is the closed halfspace determined by H ( x ) containing S d − . The conditions 0 ∈ conv X and dim(lin X ) = d ensure that P ( X ) is awell-defined, bounded polytope in R d . For any x ∈ X , let F ( x ) denote the(unique) face of P ( X ) containing x .Intuitively, the tangent polytope P ( X ) may only be large if there is abig “gap” between points of X . The following statement formalizes thisidea. Here and in the subsequent arguments, we assume that δ is smallenough so that any δ -net on S d − contains the origin in its convex hull, andis non-degenerate. Lemma 5.1.
Let X be a finite δ -net in S d − along with its tangent polytope P = P ( X ) . Then P ⊂ − δ / B d . Proof.
Let p ∈ S d − arbitrary, and let z be the intersection point of the ray op with the boundary of P . Then z = λp with some λ ≥
1. We are going toestimate | z | .Let x ∈ X be the point in X closest to p . Since X is a δ -net in S d − , wehave | p − x | ≤ δ. As before, H ( x ) is the supporting hyperplane of S d − at x . Since the dis-tance between p and H ( x ) is monotone increasing with respect to | p − x | , wenecessarily have z ∈ H ( x ). Consider the two-dimensional plane containing z, x, and o , see Figure 1. Since p ∈ H ( x ), ∠ oxz = π/ Figure 1.
We seek to find the smallest ball containing P ( X ).Denote by w the orthogonal projection of p onto the segment ox , and let | w | = t . Applying the Pythagorean Theorem for (cid:52) owp leads to | p − w | = √ − t , and for (cid:52) pwx yields(1 − t ) + 1 − t = | p − x | ≤ δ . EW ESTIMATES FOR CONVEX LAYER NUMBERS 7
Solving for t : t ≥ − δ . Finally, since (cid:52) owp ∼ (cid:52) oxz , we get that | z | = | p || w | | x | = 1 t ≤ − δ . (cid:3) We are also going to use the converse of Lemma 5.1.
Lemma 5.2.
Let X be a finite δ -net in S d − , and set P (cid:48) = conv X . Then (2) (cid:32) − δ (cid:33) B d ⊂ P (cid:48) . Proof.
Note that P (cid:48) = P ◦ , the polar body of the polytope P . Since polarityreverses containment, and ( λB d ) ◦ = (1 /λ ) B d , the statement follows. (cid:3) As usual, the inradius of a convex body in R d is the radius of the largestball contained within. Lemma 5.3.
Let X be a maximal δ -separated set in S d − , with P = P ( X ) being its tangent polytope. Then the inradius of each face of P is at least δ/ .Proof. By Lemma 2.1, X is a δ -net in S d − . Consider the spherical caps C ( x ) of S d − centered at points of x ∈ X with (Euclidean) radius δ/ C ( x ) = (cid:26) y ∈ S d − : | x − y | ≤ δ (cid:27) . Since X is a δ -separated set, the triangle inequality shows that these spher-ical caps are pairwise disjoint. Figure 2.
Estimating the inradius of F ( x )For a fixed x ∈ X , let H ( x ) be tangent hyperplane at x . Then F ( x ) ⊂ H ( x ). Let C (cid:48) ( x ) be the radial projection of C ( x ) onto H ( x ) (that is, the setof intersection points between the hyperplane H ( x ) and rays of the form oy with y ∈ C ). Since for every y ∈ C ( x ), x is the closest point to y amongpoints in X , we necessarily have that C (cid:48) ( x ) ⊂ F ( x ). On the other hand, GERGELY AMBRUS, PETER NIELSEN, AND CALEDONIA WILSON C (cid:48) ( x ) is a ( d − δ/
2. This completesthe proof. (cid:3) Construction of evenly distributed sets with large layernumbers
Starting from d = 1, we are going to construct evenly distributed familiesof point sets in B d whose layers numbers are close to the upper boundprovided by Theorem 4.1. Theorem 6.1.
For every d ≥ , there exists an evenly distributed family { X di } ∞ i =1 in R d with (3) L ( X di ) = Θ( | X di | d − d d − ) . The proof is motivated by the planar construction given in [4].
Proof.
We describe a recursive construction. For each d ≥
1, we are goingto construct a family of sets { X dn } ∞ n =1 which satisfy the following properties. • (P1): X dn ⊂ B d • (P2): For fixed d , | X dn | = Θ( n ) with the implied constant dependingon d • (P3): For fixed d, { X dn } ∞ n =1 is evenly distributed in R d • (P4): o ∈ X dn for every d and n , and it is the last remaining pointof the peeling process of X dn • (P5): L ( X dn ) = Θ( n /d − /d d − ) . We start with d = 1: for every n ≥
1, we set D n to be the set { i/n : i ∈ [ − n, n ] } . Then D n is a set of 2 n + 1 equally spaced points in [ − , n + 1. Therefore, it satisfies all the conditions (P1) – (P5) .Now, let d ≥
2, and assume that the sets X kn for every 1 ≤ k < d andevery n ≥ (P1) – (P5) . We are goingto base our construction of X dn on a parameter δ , depending on n , whosevalue we will set later. Let us fix an arbitrary n ≥
1, and construct X dn .Let D be a maximal δ -separated set in S d − with tangent polytope P = P ( D ). By Lemma 2.1,(4) | D | = Θ( δ − ( d − ) . There are | D | faces of P of the form F ( x ) with x ∈ D . Lemma 5.3 impliesthat within each face of P , a ( d − δ/ x . Set(5) m = (cid:98) δ − ( d − (cid:99) . For every x ∈ D , embed in F ( x ) a scaled copy of X d − m with scaling factor δ/ x . Denote the union of all these point sets on the faces of P with S .Let us study S . First, (4), (5) and property (P2) imply that(6) | S | = | D || X d − m | = Θ( δ − d +2 ) . Second, we refer to the fact that µ ( X d − m ) = Θ( m − / ( d − ), since X d − m isevenly distributed in R d − . Thus, the minimum distance between points of EW ESTIMATES FOR CONVEX LAYER NUMBERS 9 S contained in a given face F ( x ) is δ µ ( X d − m ) = Θ( δ ) . Because of disjointness of the ( d − δ/ P centred at the points of D , and using the scaling factor δ/
4, thedistance of a pair points of S on different faces of P is at least Ω( δ ) (cid:29) ω ( δ ).Thus,(7) µ ( S ) = Θ( δ ) . Finally, since the sets S ∩ F ( x ) are congruent to each other (and X d − m ) forevery x ∈ D , the k th layer removed by the peeling process is the union ofthe individual k th layers on the faces of P , with the last layer being the set D . Therefore, property (P5) implies that(8) L ( S ) = Θ (cid:16) m / ( d − − / ( d − d − (cid:17) = Θ (cid:16) δ − / d − (cid:17) . Next, set(9) N = (cid:22) δ (cid:23) . For every i = 1 , , . . . , N , let r i = 1 − i δ , and let(10) S i = r i S. Furthermore, let S = ∪ Ni =1 S i ∪ { o } . By (6),(11) | S | = N | S | + 1 = Θ( δ − d ) . The sets S i are to be called the shells of S . Note that r i ≥ / i ≤ N . Thus, (7) implies that the minimum distance between points in thesame shell of S is Θ( δ ).First, we show that S ⊂ B d . To that end, it is sufficient to prove that(1 − δ ) S ⊂ B d . Since S ⊂ P , Lemma 5.1 shows that(12) S = (1 − δ ) S ⊂ − δ − δ / B d ⊂ (1 − δ ) B d which is clearly contained in B d . Figure 3.
Structure of the shell constructionIn order to establish the minimum distance condition of S , we needto estimate the distance between different shells. Because of the scalingproperty, it is sufficient to give a lower bound on | x − y | with x ∈ S and y ∈ S : if | x − y | ≥ θ holds for every such pair, then | x (cid:48) − y (cid:48) | ≥ θ/ x (cid:48) , y (cid:48) contained in different shells. Clearly, | x | ≥ | y | < − δ . Thus, by the triangle inequality, | x − y | > δ , and weobtain that(13) µ ( S ) = Θ( δ ) . Next, we prove that(14) S ⊂ conv D .
Indeed, by Lemma 5.2, (cid:32) − δ (cid:33) B d ⊂ conv D .
Combined with (12), this shows that there is a ring of width δ / S and conv D .The points left by the penultimate step of the peeling process of eachshell are the scaled copies of the set D . By applying (14), we obtain thatthe peeling process on S consists of the union of the individual peelingprocesses of the shells, one after the other. In addition, the last step of thepeeling process of S is removing { o } . Thus, shells of S peel independently,and(15) L ( S ) = N L ( S ) + 1 = Θ (cid:16) δ − / d − (cid:17) , by (8) and (9).Let us now set δ = n − / (2 d ) . EW ESTIMATES FOR CONVEX LAYER NUMBERS 11
Then by (11), S is a point set with Θ( n ) points, contained in B d , whichshows (P1) and (P2) . By (13), the minimum distance condition µ ( S ) =Θ( n − /d ) holds, implying (P3) . The origin o is contained in S , and it is thelast point when peeling S , which ensures that (P4) holds. Finally, by (15), L ( S ) = Θ (cid:16) n /d − /d d − (cid:17) , which agrees with (P5) . Thus, taking X dn = S , the resulting family of setssatisfies all the conditions (P1) – (P5) . This concludes the proof. (cid:3) We remark that by defining L d so that L ( X dn ) = Θ( n L d ) in the aboverecursive construction, we obtain the following recurrence relation for L d :(16) 2 dL d = 2 + ( d − L d − with L = 1. This may be solved by using exponential generating functions.Setting F ( x ) = ∞ (cid:88) k =1 x k L k , (16) leads to F (cid:48) ( x ) = 22 − x + x . This, in turn, shows that L d = 2 d − d d , leading to the exponent in (3).We also note that for d = 2, a different construction may be given with thesame order of magnitude for the layer number. The construction consistsof n / regular n / -gons placed in a spiralling, interlocking manner (seeFig. 6). Given one layer, the next is obtained by moving n − / distance oneach side of the polygon in positive orientation. Such a construction in R gives a weaker bound than (3). Figure 4.
Spiral construction in the plane
Finally we remark that the existence of an evenly distributed family ofpoint sets X in R satisfying L ( X ) = Θ( | X | ) would also lead to a construc-tion with L ( X ) = Θ( | X | d ) for all d ≥
3. Determining the exact asymptoticsfor the maximal layer number of an evenly distributed family remains anopen question at this point.7.
Acknowledgements
This research was done under the auspices of the Budapest Semesters inMathematics program. We are grateful to W. Joo for communicating theresults of [4] to us.
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SIAM J. Discrete Math. (2013),no. 2, 650–655.[6] W. Joo, The layer number of evenly distributed point sets . MA Thesis, KAIST, 2016.[7] H. Raynaud,
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E-mail address , G. Ambrus: [email protected]
Peter Nielsen, Department of Mathematics, University of Wisconsin-Madison,480 Lincoln Dr, Madison, WI 53706, U.S.A.
E-mail address , P. Nielsen: [email protected]
Caledonia Wilson, Mount Holyoke College, 50 College St, MA 01075,U.S.A.
E-mail address , C. Wilson:, C. Wilson: