aa r X i v : . [ m a t h . M G ] F e b The right acute angles problem?
Andrey Kupavskii ∗ , Dmitriy Zakharov † Abstract
The Danzer–Gr¨unbaum acute angles problem asks for the largest size of a set ofpoints in R d that determines only acute angles. There has been a lot of progressrecently due to the results of the second author and of Gerencs´er and Harangi, andnow the problem is essentially solved.In this note, we suggest the following variant of the problem, which is one wayto “save” the problem. Let F ( α ) = lim d →∞ f ( d, α ) /d , where f ( d, α ) is the largestnumber of points in R d with no angle greater than or equal to α . Then the question isto find c := lim α → π/ − F ( α ) . It is an intriguing question whether c is equal to 2 as onemay expect in view of the result of Gerencs´er and Harangi. In this paper we prove thelower bound c > √ A set of points X ⊂ R d is called acute ( non-obtuse ) if any three points from X form anacute (acute or right, respectively) triangle. In 1962, Danzer and Gr¨unbaum [DG] confirmeda conjecture of Erd˝os from 1957 that any non-obtuse set of points in R d has cardinality atmost 2 d , moreover, the only examples of non-obtuse sets of cardinality 2 d are the hypercubeand some of its affine images. They then modified the question and asked to determine themaximum size f ( d ) of an acute set in R d for any d >
2. Danzer and Gr¨unbaum obtainedthe first bounds on f ( d ): 2 d − f ( d ) d − , (1)where the upper bound immediately follows from the aforementioned result on non-obtusesets. They conjectured that the lower bound is tight.As it turned out recently, the value of f ( d ) is actually very close to the upper bound in(1). While the only improvement upon the upper bound in (1) made so far is the inequality f (3) ∗ Moscow Institute of Physics and Technology, IAS Princeton ; Email: [email protected]
Researchsupported by the grant of the Russian Government N 075-15-2019-1926. † Higher School of Economics, Email: s18b1 [email protected] f ( d ) that are known at the moment are f (2) = 3 and f (3) = 5, and the latter isthe only known improvement of the upper bound (1), due to Croft [C].In 1983, Erd˝os and F¨uredi [EF] provided a probabilistic construction of an acute set with[ ( √ ) d ] points, thus disproving the conjecture of Danzer and Gr¨unbaum. The underlyingidea was to consider a random subset of the vertices of the hypercube { , } d (see the nextsection for details). In the years 1983-2009, the improvements of the lower bound were verymoderate: the constant in front of the exponential term ( √ ) d was improved in severalsteps, resulting in the inequality f ( d ) & . · ( √ ) d [B, Bu]. In 2009, Ackerman and Ben-Zwi [AB] improved the Erd˝os–F¨uredi bound by a factor of c √ d using a certain general resultconcerning the independence numbers of sparse hypergraphs. In 2001, Harangi [H] made thefirst exponential improvement: the constant √ ≈ .
155 was replaced by ( ) . ≈ . X n ⊂ R d n , rather than { , } d , as it was done in the proof by Erd˝os and F¨uredi. Here, X ⊂ R d is a low-dimensionalacute set, which is typically constructed by hand or with the help of computer. For example,if one takes X to be an acute triangle on the plane then one gets the bound f ( d ) & . d , which is slightly better than the Erd˝os–F¨uredi bound. Harangi used a 12-point acute subsetof R in his proof.The next round of development was triggered in the spring of 2017, when the first explicitexponential acute sets were constructed by the second author [Z]. The obtained bound on f ( d ) was also much better than the previously known ones: f ( d ) > F d +1 ≈ . d , where F d is the d -th Fibonacci number. The proof used induction and certain slight perturbationsof the point set to make the right angles in the arising product-type constructions acute. Inthe fall of 2017 Gerencs´er and Harangi [GH] proved that f ( d ) > d − + 1 . (2)The proof was inspired by constructions of 9-point and 17-point acute sets in R and R ,respectively, made by an Ukranian mathematics enthusiast. The idea of Gerencs´er andHarangi’s bound is to carefully perturb the vertices of the hypercube { , } d − using oneextra dimension to get rid of all right angles. One extra point can then be added to theconstruction.One common feature of all known explicit exponential-sized constructions is that thelargest angle among the points is just barely smaller than π , and the constructions breakdown completely if we require the largest angle to be, say π − . π . This suggests a certain interesting direction for research, butlet us first introduce a couple of definitions. Definition 1.
Denote by f ( d, α ) the size of the largest set of points in R d with no threepoints forming an angle at least α . Put F ( α ) := lim sup d →∞ f ( d, α ) /d . (3) Here F = F = 1. f ( d ) = f ( d, π ), and the result of Gerencs´er–Harangi now implies that F ( π ) = 2. In [Kup], the first author showed that lim α → π/ + f ( d, α ) = 2 d .Note that f ( d, α ) is meaningful only for α ∈ [ π , π ] since f ( d, α ) = 2 for any α π . Somefurther results about f ( d, α ) for α close to π or to π can be found in [EF].Results of Erd˝os–F¨uredi [EF, Theorem 3.6] translate to the following: F (cid:0) π δ (cid:1) ∈ (cid:2) δ , δ (cid:3) . (4)In the range α > π it turns out that f ( d, α ) grows surprisingly fast. The following resultis essentially due to Erd˝os–F¨uredi [EF, Theorem 4.3] but their formulation applies only to α close enough to π (note that the condition that n is sufficiently large is missing in thestatement of [EF, Theorem 4.3]). Proposition 1.
For any α ∈ ( π , π ) there are constants C, c > such that for all sufficientlylarge d c d < f ( d, α ) < C d . (5)Note that Proposition 1 refutes Conjecture 2.13 from the very same paper [EF].Now we can formulate our main question. Question 1.
Is it true that lim α → π/ − F ( α ) = 2? (6) Equivalently, is it true that for any ε > there is δ > so that for any sufficiently large d there is a set X ⊂ R d of cardinality at least (2 − ε ) d such that any three points from X determine an angle less than π − δ ? Although the problem is very close to the acute angles problem, the current methods thatuse explicit constructions fail completely, and the gap between the bounds is still exponential.We prove the following lower bound in this paper.
Theorem 1.
We have lim α → π/ − F ( α ) > √ . (7) That is for every ε > there exists δ > such that for any sufficiently large d there is a set X ⊂ R d of cardinality at least ( √ − ε ) d determining only angles less than π − δ . Our proof is a combination of the method of Erd˝os–F¨uredi with the recent constructionof acute sets by Gerencs´er–Harangi.The second result gives a non-trivial upper bound on F ( α ) for any α < π/ Theorem 2.
For α > small enough we have F ( π − α ) − α . f ( d ) d due to Danzer and Gr¨unbaum.Namely, their proof is based on the observation that if X is an acute set and P = conv( X )is the convex hull of X then interiors of homothets P + x , x ∈ X , are pairwise disjoint.Considering the volumes one easily obtains the bound | X | d . The idea behind theproof of Theorem 2 is to take two disjoint subsets A, C ⊂ X and consider sets of the form λ conv( A ) + (1 − λ ) c ⊂ conv( A ∪ C ), where c ∈ C . One can show that these sets are pairwisedisjoint provided (i) all the angles in X are less than π − α and (ii) λ is chosen appropriately.One then obtains an inequality λ d Vol(conv A ) | C | Vol(conv A ∪ C ). Lemma 1 implies thatone can choose A and C in such a way that Vol(conv A ) and Vol(conv A ∪ C ) are almost thesame and | C | is comparable to | X | , which completes the proof. Sketch of the proof of Proposition 1.
To prove the lower bound, we construct a set { v , . . . , v m } of m > c d unit vectors in R d such that the angle between any two of them lies in ( π − ε, π + ε ),where 2 ε = α − π . This can be done by taking a random subset on the unit sphere andapplying a concentration inequality (see, for instance, [M, Chapter 14]). Now take a suffi-ciently large number λ and consider the set X = { v I = P t ∈ I λ t v t | I ⊂ [ m ] } . Note that | X | = 2 c d . For any two points v I , v J ∈ X we have v I − v J ≈ ± λ t v t , where t is the largestelement of I ∆ J . So the angle between v I − v J and v I − v K is approximately equal to theangle between some vectors ± v i and ± v j , and therefore, it is at most α .To prove the upper bound, we construct a set { v , . . . , v m } of m C d vectors such thatany vector determines an angle less than π − α with one of them. This can be done by agreedy algorithm or deduced from known results for the sphere packing problem. Take aset X of more than 2 m points. For x, y ∈ X , color a pair ( x, y ) , x = y , in color i if theangle between v i and x − y is at most π − α . In what follows, we show that, since | X | > m ,there exists a triple x, y, z such that ( x, y ) and ( y, z ) received the same color (i.e., there is amonochromatic oriented 2-path). But then the angle between y − x and y − z is at least α .We show that such a triple exists by induction on m . The statement is clear for m = 1and | X | = 3. Next, for m -colorings, take any color, say, red, and consider all edges of thiscolor. If there is no red oriented 2-path, then each vertex either has only incoming or onlyoutgoing red edges, and so red edges span a bipartite graph. (We are free to assign verticeswith no incident red edge to any of the two parts.) Take the bigger part of this bipartitegraph. It has size at least ⌈ (2 m + 1) / ⌉ = 2 m − + 1 and is colored with m − Proof of Theorem 1.
Fix an arbitrary ε >
0. Take a sufficiently large d and an acute set X ⊂ R d of size 2 d − + 1 (which exists by (2)). Let R > X and denoteby s the smallest scalar product h x − y, x − z i over all triples x, y, z ∈ X such that x = y, z .By the definition of an acute set, we have s > d divides d . Let m = 2 − ε nd where n = d/d . Choose 2 m uniformly random points p , . . . , p m ∈ X n ⊂ R d n , and set p i = ( p i , . . . , p in ). Let usestimate the expectation of the number of triples ( i, j, k ) such that h p i − p j , p i − p k i ε ns .If for some i, j, k we have h p i − p j , p i − p k i ε ns then there are at least (1 − ε ) n coordinates t ∈ { , . . . , n } for which p it = p jt or p it = p kt . The probability of the latter event is at most (cid:0) n ε n (cid:1) (cid:16) | X | (cid:17) (1 − ε ) n n − (1 − ε )( d − n . So the expectation of the number of such triples is atmost (2 m ) n − (1 − ε )( d − n m (1 − ε ) nd − (1 − ε ) nd +3 n ≪ m. (8)Thus there are points p , . . . , p m with at most m “bad” triples. Remove one point from eachof these triples and obtain a set X ⊂ X n ⊂ R nd of cardinality at least m = √ (1 − ε ) nd suchthat for any two points x, y ∈ X we have | x − y | R n and for any three points x, y, z ∈ X we have h x − y, x − z i > ε ns . This means that the angle α between vectors x − y, x − z satisfies cos α > ε s/R and thus depends on ε only.In the proof of Theorem 2, we shall need the following lemma. Lemma 1.
Suppose X ⊂ R d , | X | = N > d + 1 and the convex hull conv( X ) has non-zerovolume. Then for any c ∈ [ d log NN , there are sets A ⊂ B ⊂ X such that1. | B \ A | > c d log N N .2. = Vol(conv( B )) (1 + c )Vol(conv( A )) .Proof. By Carath´eodory’s theorem, every point of conv( X ) lies in the convex hull of some d + 1 points of X , so by the pigeonhole principle, there is a set X ⊂ X of size d + 1 suchthat Vol(conv( X )) > (cid:18) Nd + 1 (cid:19) − Vol(conv( X )) > N − d − Vol(conv( X )) . Take any chain X ⊂ X ⊂ . . . ⊂ X m = X, such that | X i +1 \ X i | ∈ [ c d log N N, c d log N N ](it is possible because of the restriction on c ). We have m > d log Nc , so if we hadVol(conv( X i +1 )) > (1 + c )Vol(conv( X i )) for all i , thenVol(conv( X )) > (1 + c ) m Vol(conv( X )) > d log N Vol(conv( X )) > Vol(conv( X )) , a contradiction. Proof of Theorem 2.
Take a set X ⊂ R d which determines only angles at most π − α for asufficiently small α >
0. Put ε = sin α . It is easy to see that for any three different points x, y, z ∈ X h y − x, z − x i > ε k y − x kk z − x k > . ε k z − x k , (9)where the last inequality follows from the fact that k y − x kk z − x k = sin ∠ xzy sin ∠ zyx > sin ∠ xzy > sin 2 α > . ε for sufficiently small α . Doing the same calculation for both z − x and x − z as thesecond vector in the scalar product in (9), we get that for any three distinct x, y, z we have1 . ε k z − x k < h y − x, z − x i < (1 − . ε ) k z − x k . (10)5pplying Lemma 1 with c = 1 we get sets A ⊂ B such that 0 = Vol(conv B ) A ) and | B \ A | > | X | d . Take λ = · (1 − . ε ) − , from (10) we see that for any dis-tinct x, z ∈ B \ A we have ((1 − λ ) x + conv( λA )) ∩ ((1 − λ ) z + conv( λA )) = ∅ . Indeed, for anypoint y from the first set we have h y − x, z − x i < λ (1 − . ε ) k z − x k = k z − x k , while for any y ′ from the second set we have h y ′ − x, z − x i > (1 − λ ) k z − x k + λ · . ε k z − x k = k z − x k .Moreover, (1 − λ ) x + conv( λA ) ⊂ conv B for any x ∈ B , so | B \ A | λ d Vol(conv A ) Vol(conv B ) A ) , (11)thus | X | d | B \ A | d λ − d = 8 d d (cid:0) − . ε (cid:1) d (2 − α ) d , (12)provided that d is sufficiently large and α > α → + sin αα = 1.) Acknowledgements:
We thank the reviewers for carefully reading the manuscriptand suggesting numerous changes that helped to improve the exposition.
References [AB] E. Ackerman and O. Ben-Zwi,
On sets of points that determine only acute angles ,European Journal of Combinatorics 30 (2009), N4, 908–910.[B] D. Bevan,
Sets of points determining only acute angles and some related colouring prob-lems , the electronic journal of combinatorics 13 (2006), N1, paper 12.[Bu] L. V. Buchok,
Two New Approaches to Obtaining Estimates in the Danzer–Gr¨unbaumProblem , Math. Notes, 87 (2010), N4, 489–496.[C] H. T. Croft,
On 6-Point Configurations in 3-Space,
Journal of the London MathematicalSociety 1 (1961), N1, 289–306.[DG] L. Danzer and B. Gr¨unbaum, ” ¨Uber zwei Probleme bez¨uglich konvexer K¨orper von P.Erd˝os und von VL Klee , Mathematische Zeitschrift 79 (1962), N1, 95–99.[EF] P. Erd˝os and Z. F¨uredi,
The greatest angle among n points in the d-dimensional Eu-clidean space , Annals of Discrete Mathematics 17 (1983), 275–283.[GH] B. Gerencs´er and V. Harangi,
Acute sets of exponentially optimal size , Discrete &Computational Geometry 62 (2019), N4, 775–780.[H] V. Harangi,
Acute sets in Euclidean spaces , SIAM Journal on Discrete Mathematics 25(2011), N3, 1212–1229.[M] Matouˇsek, Jiˇr´ı.
Lectures on discrete geometry . Vol. 212. New York: Springer, 2002.6Kup] A. Kupavskii,
Number of double-normal pairs in space , Discrete and ComputationalGeometry 56 (2016), N3, 711–726.[Z] D. Zakharov,