Expected mean width of the randomized integer convex hull
aa r X i v : . [ m a t h . M G ] M a r Expected mean width of the randomizedinteger convex hull
Hong Ngoc Binh, Matthias Reitzner
Abstract
Let K ∈ R d be a convex body, and assume that L is a randomly ro-tated and shifted integer lattice. Let K L be the convex hull of the (ran-dom) points K ∩ L . The mean width W ( K L ) of K L is investigated. Theasymptotic order of the mean width difference W ( λK ) − W (( λK ) L )is maximized by the order obtained by polytopes and minimized bythe order for smooth convex sets as λ → ∞ . Let K be a convex body and Z d the integer lattice in R d . The convex hull[ K ∩ Z d ] of the intersection of K with Z d yields a polytope K Z d , the integerconvex hull of K . The higher dimensional Gauss circle problem asks for K = λB d , the ball of radius λ >
0, how many integer points are containedin K Z d compared to its volume V d ( K ). The metric variant we consider herecompares the volume of K Z d to the volume of K , and more generally theintrinsic volumes V j ( K Z d ) to the intrinsic volumes of K . This problem has along history, and more recent investigations have been motivated by questionsfrom integer programming and enumeration problems. We refer to the articleby B´ar´any and Larman [3] and the survey article by B´ar´any [2] for moredetails.It is immediate that all these problems depend on the position, size andshape of K in a delicate way. Consider e.g. the enlarged unit cube K = λC d = [ − λ, λ ] d , λ >
0, where all functionals of K Z d are locally constant for λ / ∈ N and have jumps at λ ∈ N . This is due to the fact that C d is in aspecial position with respect to Z d . Therefore it is of interest to ask whathappens in generic situations.This question was made precise by B´ar´any and Matouˇsek [4] who inves-tigated the integer convex hull of λK when K is in a random position, i.e.1 is a randomly rotated and shifted copy of a convex body K . Alterna-tively, one can intersect K with a random lattice L , a randomly shiftedand rotated copy of Z d , which yields the randomized integer convex hull, K L = [ K ∩ L ].Of interest are metric quantities of this random polytope likethe volume, surface area, mean width, and combinatorial quantities like thenumber of faces.This problem turns out to be surprisingly difficult even in simple cases.B´ar´any and Matouˇsek proved that the expected number of vertices of K L is connected to the so-called floating body of K , if the boundary of K issufficiently smooth. Further, in the planar case, they proved integral boundsfor the expected area difference V ( λK ) − V (( λK ) L ) which led to the bounds c (ln V ( λK )) | {z } ≈ (ln λ ) ≤ V ( λK ) − E V (( λK ) L ) ≤ c V ( λK ) | {z } ≈ λ (1)for λ sufficiently large. The lower bound is attained for polygons, and theupper bound for smooth convex sets. We are not aware of any other resultson the random integer convex hull K L .Surprisingly the behaviour in formula (1) changes if we consider the meanwidth instead of the area. To define the mean width, consider for given u ∈ S d − two parallel hyperplanes orthogonal to u squeezing K . The distancebetween these two hyperplanes is the width W ( K, u ) in direction u (for aformal definition see Section 2). The mean width is given by W ( K ) = 1 ω d Z S d − W ( K, u ) du. Up to a constant, the mean width is in the planar case the perimeter P ( K )of a convex body K , and in general dimensions the first intrinsic volume (forthe definition of intrinsic volume we refer to Section 4).The first main result of our paper gives upper and lower bounds on theexpected mean width difference. Theorem 1.1.
Let K be an arbitrary convex body. Then there are constants γ ( K ) , γ ( K ) such that γ ( K ) λ − d − d +1 ≤ W ( λK ) − E ( W (( λK ) L )) ≤ γ ( K ) as long as λ ≥ λ ( K ) . γ ( K ) P ( λK ) − | {z } ≈ λ − ≤ P ( λK ) − E P (( λK ) L ) ≤ γ ( K ) . Note that the upper bound of Theorem 1.1 can be generalized to allintrinsic volumes.
Corollary 1.2.
Let K be a convex body. Then there is a constants γ ( K ) ,such that V j ( λK ) − E ( V j (( λK ) L )) ≤ γ ( K ) V j − ( λK ) for all j ∈ { , . . . , d } , as long as λ ≥ λ ( K ) . Yet it is clear from (1) that this inequality is not optimal in general.The area difference is maximized by smooth convex sets, in contrast to themean width difference which is maximized by polytopes. It would be of highinterest to generalize the results of B´ar´any and Matouˇsek (1) and Theorem1.1 to sharp inequalities for all intrinsic volumes.Theorem 1.1 concerning the mean width is optimal as shown by polytopesand smooth convex sets.
Theorem 1.3.
Let P be a polytope. Then there is a constant γ ( P ) > suchthat lim λ →∞ W ( λP ) − E ( W (( λP ) L )) = γ ( P ) . Theorem 1.4.
Assume K is a smooth convex body. Then there is a constant γ ( K ) such that W ( λK ) − E ( W (( λK ) L )) ≤ γ ( K ) λ − d − d +1 for λ sufficiently large. Thus in the planar case and for smooth convex bodies, the mean widthdifference is of order λ − which tends to zero, and by the result of B´ar´anyand Matouˇsek [4] the volume difference is of order λ which tends to infinity.To compare the two results heuristically, one should check that the volumedifference is approximately the perimeter P ( λK ) times the mean distance of λK and ( λK ) L , which is the mean width, V ( λK ) − E V (( λK ) L ) ≈ P ( λK ) ( W ( λK ) − W (( λK ) L )) ≈ λ λ − = λ We work in d -dimensional Euclidean space R d with inner product h· , ·i , anddenote by B d its unit ball and by S d − = ∂B d the unit sphere. Here ∂K is the boundary of a set K ⊂ R d . By B ( x, r ) we denote a ball with center x and radius r . The volume of B d is κ d , and the spherical Lebesgue- orHausdorff-measure of S d − is ω d = dκ d .Let L be the set of rotated and translated integer lattices in R d , L = (cid:8) L t,ρ = ρ ( Z d + t ) : t ∈ [0 , d , ρ ∈ SO ( d ) (cid:9) . For A ⊂ R d we write R A f ( x ) dx for integration with respect to the d -dimensional Lebesgue measure, and analogously R A f ( u ) du for integrationwith respect to spherical Lebesgue measure for A ⊂ S d − , R A f ( ρ ) dρ forintegration with respect to the Haar probability measure for A ⊂ SO d , and Z L f ( L ) dL = Z [0 , d Z SO d f ( L t,ρ ) dρ dt. Thus the ‘uniform measure’ on L is given by uniformly chosen t ∈ [0 , d and ρ ∈ SO d , and for a set A ⊂ L we have P ( L ∈ A ) = Z L ( L ∈ A ) dL, E f ( L ) = Z L f ( L ) dL , where ( · ) denotes the indicator function.The convex hull of a set A is denoted by [ A ], K d denotes the set of convexbodies, i.e. compact convex sets with nonempty interior, P d ⊂ K d the set ofconvex polytopes. For given K ⊂ K d , the randomized integer convex hull isthe random polytope defined by K L := [ K ∩ L ] , where L ∈ L is chosen uniformly. 4e are interested in the distance between K and K L . To define thedistance let h K ( u ) = max {h x, u i : x ∈ K } be the support function of K ∈ K d in direction u ∈ S d − . The Hausdorffdistance d H ( K, Q ) between two convex bodies
K, Q is given by d H ( K, Q ) = max u ∈ S d − | h K ( u ) − h Q ( u ) | . (2)Note that H K ( u ) = { x ∈ R d : h x, u i ≥ h K ( u ) } is a supporting halfspaceto K with unit normal vector u . For each u ∈ S d − and t ∈ [0; + ∞ ), wedenote by K t,u the cap of width t cut off from K by a halfspace parallel to H K ( u ), K t,u = { x ∈ K : h x, u i ≥ h K ( u ) − t } . For u ∈ S d − , the width of K in direction u is defined by W ( K, u ) = h K ( u ) + h K ( − u ) , and the mean width of K is W ( K ) = 1 ω d Z S d − W ( K, u ) du = 2 ω d Z S d − h K ( u ) du. We are interested in the distance between K and K L measured in terms of thedifference of the mean width W ( K ) − W ( K L ). Observe that since K L ⊂ K this difference is always postive and equals zero if and only if K = K L . Westart with a simple but crucial lemma. Lemma 3.1.
Assume that K ∈ K d . Then W ( K ) − E ( W ( K L )) = 2 ω d Z S d − ∞ Z P ( K t,u ∩ L = ∅ ) dtdu. Proof.
The difference of the expected mean width of K and K L is by defini-tion W ( K ) − E ( W ( K L )) = 2 ω d E Z S d − ( h K ( u ) − h K L ( u )) du. K L ⊂ K , the integrand is postive, and Fubinis theorem yields W ( K ) − E ( W ( K L )) = 2 ω d E Z S d − ∞ Z ( t ≤ h K ( u ) − h K L ( u )) dtdu = 2 ω d E Z S d − ∞ Z ( K t,u ∩ L = ∅ ) dtdu = 2 ω d Z S d − ∞ Z P ( K t,u ∩ L = ∅ ) dtdu. (3)Hence estimating the mean width difference boils down to estimate theprobability that a cap avoids the random lattice L . The following upperbound was stated by B´ar´any and Matouˇsek [4] and proved by B´ar´any [1]. Lemma 3.2.
There exist constants ν > and c > (both depending on d )such that for every convex body K ∈ K d with V d ( K ) ≥ ν , P ( K ∩ L = ∅ ) ≤ cV d ( K ) holds. We give a simple lower bound which turns out to have the right order inthe applications we need in this work.
Lemma 3.3.
For any measurable set A ⊂ R d we have P ( A ∩ L = ∅ ) ≥ − V d ( A ) . Proof.
We start by calculating the expected number of lattice points in A . E ( { A ∩ L } ) = Z L X z ∈ L A ( z ) dL = Z SO d Z [0 , d X ω ∈ Z d A ( ρ ( ω + t )) dtdρ = Z SO d X ω ∈ Z d Z [0 , d + ω A ( ρ ( y )) dydρ = Z SO d Z R d A ( ρ ( y )) dydρ = Z SO d Z R d A ( x ) dxdρ = V d ( A )6herefore, V d ( A ) = E ( { A ∩ L } ) = ∞ X i =1 i P ( { A ∩ L } = i ) ≥ ∞ X i =1 P ( { A ∩ L } = i )= P ( A ∩ L = ∅ ) , and hence, P ( A ∩ L = ∅ ) = 1 − P ( A ∩ L = ∅ ) ≥ − V d ( A ). In this section we prove bounds for general convex bodies. We start withthe upper bound. The following lemma is somehow connected to Khintchin’sFlatness Theorem [6], see also [7]. It states that a cap which is too fat cannotavoid any lattice.
Lemma 4.1.
Let K be a convex body. Then there are constants τ ( K ) , λ ( K ) such that for t ≥ τ ( K ) and λ ≥ λ ( K ) we have ( λK ) t,u ∩ L = ∅ for all u ∈ S d − and all L ∈ L .Proof. Assume w.l.o.g. that the inball E of K is centered at the origin, anddenote by x ( u ) ∈ ∂K a boundary point with outer unit normal vector u .Then K contains the cone C u = [ x ( u ) , E ∩ u ⊥ ]with base E ∩ u ⊥ and apex x ( u ). Denote by r ( u ) the radius of the inball of C u . Thus sr ( u ) is the inball of sC u . Observe that any ball of radius at least √ d meets any lattice L = ρ ( Z d + t ). Thus for s = √ d r ( u ) (4)the cone sC u must contain a lattice point. The essential observation is that sC u is a cone with height sh K ( u ), and that ( λC u ) t,u also is a homothetic copyof C u with height t . Hence (4) implies( λC u ) t,u ∩ L = (cid:18) th K ( u ) C u (cid:19) ∩ L = ∅ t ≥ sh K ( u ) = √ d h K ( u )2 r ( u )as long as λ ≥ s = √ d r ( u ) . We define τ ( K ) := max u ∈ S d − √ d h K ( u )2 r ( u ) , and λ ( K ) := max u ∈ S d − √ d r ( u )and obtain for t ≥ τ ( K ) and λ ≥ λ ( K )( λC u ) t,u ∩ L = ∅ for all u ∈ S d − and L ∈ L . Since ( λC u ) t,u ⊂ ( λK ) t,u this yields the lemma.There are some immediate consequences. By Lemma 4.1, for each u ∈ S d − the distance of the support functions of K and K L is at most τ ( K ),which by definition gives a simple upper bound for the Hausorff distance (2)and for the mean width difference (3). Putting γ ( K ) = 2 τ ( K ) this is thestated upper bound in Theorem 1.1. Theorem 4.2.
Let K be a convex body. Then there is a constant τ ( K ) suchthat for λ sufficiently large d H ( λK, ( λK ) L ) ≤ τ ( K ) (5) for any lattice L ∈ L , and W ( λK ) − E ( W (( λK ) L )) ≤ τ ( K ) . The intrinsic volumes V j ( K ) of a convex body K , j = 0 , . . . , d , are definedas the coefficients in the Steiner formula, V d ( K + εB d ) = d X i =0 κ i V d − i ( K ) ε i , where e.g. 2 V d − ( K ) is the surface area of K , κ d − dκ d V ( K ) equals the meanwidth W ( K ), and V ( K ) = 1 is the Euler characteristic of K . By Kubotasformula, the intrinsic volumes of a convex body can be written as V j ( K ) = c − d,k,j Z G dk V j ( K | G ) dG, = 0 , . . . , k , where G dk is the Grassmann manifold of the k -dimensional sub-spaces of R d , integration is with respect to the Haar probability measure on G dk , and c d,k,j = k !( d − j )! κ d − j κ k d !( k − j )! κ d κ k − j . Because of (5), λK ⊂ ( λK ) L + τ ( K ) B d , and this also holds for all projectionsonto k -dimensional subspaces. Hence inequality (5) implies V j ( λK | G ) − V j (( λK ) L | G ) ≤ τ ( K ) 2 V j − ( λK | G ) , and Kubotas formula yields an upper bound for the intrinsic volumes. Corollary 4.3.
Let K be a convex body. Then there is a constant τ ( K ) ,such that for λ sufficiently large V j ( λK ) − E ( V j (( λK ) L )) ≤ c d,j − ,j c d,j,j τ ( K ) V j − ( λK ) for all j ∈ { , . . . , d } . For a general lower bound on the mean width we need the followingLemma. It is a dual version of Blaschke’s rolling theorem, and closely relatedto results of McMullen [8] and Sch¨utt and Werner [9] for balls rolling insidea convex body. The dual version could be deduced from these results usinga duality argument, and is stated explicitly in a paper by B¨or¨oczky, Fodorand Hug [5].
Lemma 4.4 ([5], Lemma 5.2 ) . Let K ∈ K d be a convex body. There exists ameasurable set Σ ⊂ S d − with positive spherical Lebesgue measure, and some R > , all depending on K , such that for any u ∈ Σ there is some p ∈ ∂K such that K ⊂ p + R ( B d − u ) . The next theorem states the lower bound from Theorem 1.1.
Theorem 4.5.
Assume K ⊂ B d is a convex body. Then there is a constant γ ( K ) , such that W ( λK ) − E ( W (( λK ) L )) ≥ γ ( K ) λ − d − d +1 for λ ≥ . We prepare the proof of this theorem by the following lemma.9 emma 4.6.
Let B (0 , r ) be a ball of radius r . Then c r d − t d +12 ≤ V d ( B (0 , r ) t,u ) ≤ c r d − t d +12 . Proof.
For r = 1, the intersection of B d with a hyperplane of distance 1 − t from the origin is a ( d − √ t − t ∈ [ t, t ] . The volume of the cap B dt,u is bounded from above by the volume of a cylinderand from below by the volme of a cone whose base are the same ( d − d κ d − t d +12 ≤ V d ( B dt,u ) ≤ d − κ d − t d +12 . Because V d ( B (0 , r ) t,u ) = r d V d ( B dt/r,u )this proves the lemma. Proof of Theorem 4.5.
We substitute t = λ − d − d +1 x and obtain λ d − d +1 ( W ( λK ) − E ( W (( λK ) L )))= 2 ω d Z S d − ∞ Z λ d − d +1 P (( λK ) t,u ∩ L = ∅ ) dtdu = 2 ω d Z S d − ∞ Z P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) dxdu (6)By Lemma 4.4 there exists a suitable set Σ ⊂ S d − with λ d − (Σ) > R > K ⊂ x + R ( B d − u ) . For u ∈ Σ, by Lemma 3.3 and Lemma 4.1 we have the lower bound P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) ≥ − V d (cid:18) ( λK ) λ − d − d +1 x,u (cid:19) ≥ − V d (cid:18) B (0 , λR ) λ − d − d +1 x,u (cid:19) where we used that λK is contained in a ball of radius λR . Because ofLemma 4.6, we have P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) ≥ − c R d − x d +12 . λ d − d +1 ( W ( λK ) − E ( W (( λK ) L ))) ≥ ω d Z Σ ∞ Z P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) dxdu ≥ ω d Z Σ du ∞ Z (cid:16) − c R d − x d +12 (cid:17) + dx == γ ( K )where the constant γ ( K ) depends on Σ and R and thus on K . In this section we prove a precise version of the upper bound in Theorem 1.4concerning smooth convex bodies. Fix the dimension d ≥
2, and for r > K ( r ) the set of convex bodies where a ball of radius r rolls inside K , i.e. K ∈ K d and for all p ∈ ∂K there exist a unit vector u ∈ S d − with p + r ( B d − u ) ⊂ K. Theorem 5.1.
Assume K ∈ K ( r ) . Then there is a constant γ dependingon r , such that W ( λK ) − E ( W (( λK ) L )) ≤ γ λ − d − d +1 for λ sufficiently large.Proof. To prepare for the use of Lemma 3.2 in the following, we assume that λ ≥ λ ( r ) where λ ( r ) is chosen such that V d ( B (0 , λ ( r ) r ) = 2 ν. As in the proof of Theorem 4.5 we start with λ d − d +1 ( W ( λK ) − E ( W (( λK ) L )))= 2 ω d Z S d − ∞ Z P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) dxdu.
11e first use that each boundary point of λK is touched from inside by aball of radius λr , and then Lemma 3.2, λ d − d +1 ( W ( λK ) − E ( W (( λK ) L )))= 2 ω d Z S d − ∞ Z P (cid:18) B (0 , λr ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) dxdu ≤ x Z dx + 2 ∞ Z x cV d (cid:18) B (0 , λr ) λ − d − d +1 x,u (cid:19) dx. (7)Here x = x ( r ) is chosen such that V d (cid:18) B (0 , λr ) λ − d − d +1 x ,u (cid:19) = ν, which by Lemma 4.6 implies (cid:18) νc (cid:19) d +1 r − d − d +1 ≤ x ≤ (cid:18) νc (cid:19) d +1 r − d − d +1 . The first integral in (7) is bounded by 2 x .For the second integral in (7) we use Lemma 4.6 to obtain2 ∞ Z x cV d (cid:18) B (0 , λr ) λ − d − d +1 x,u (cid:19) dx ≤ cc r − d − ∞ Z x x − d +12 dx = 4 cc ( d − r − d − x − d − . Therefore, λ d − d +1 ( W ( λK ) − E ( W (( λK ) L ))) ≤ γ where γ depends on r .It would be helpful, if for smooth K ∈ K d we have the convergence P (cid:18) ( λK ) λ − d − d +1 x,u ∩ L = ∅ (cid:19) → f K ( x, u )as λ → ∞ , with some measurable function f K ( x, u ). Yet we have not beenable to prove that. 12 Polytopes
The preceding section shows that the lower bound in Theorem 1.1 cannot beimproved in general since it is - up to constants - sharp for smooth convexbodies. In this section we prove that also the upper bound is optimal - upto constants.
Theorem 6.1.
Let P ∈ P d be a d -dimensional polytope with nonempty in-terior. Then there is a constant γ ( P ) > such that lim λ →∞ W ( λP ) − E ( W (( λP ) L )) = γ ( P ) . Proof.
The polytope P is the convex hull of its vertices v ∈ F ( P ), and foreach vertex v we denote by N ( v ) ⊂ S d − the (relatively open) normal coneat v , i.e. the set of all unit vectors orthogonal to a supporting hyperplane to P touching P at v . N ( v ) = { u ∈ S d − : H K ( u ) ∩ P = v } . Since P is a polytope, the set of unit vectors in S d − \ [ v ∈F ( P ) N ( v ) is a null set with respect to spherical Lebesgue measure. Thus Lemma 3.1gives lim λ →∞ (cid:16) W ( λP ) − E W (( λP ) L ) (cid:17) == X v ∈F ( P ) lim λ →∞ Z N ( v ) ∞ Z P (( λP ) t,u ∩ L = ∅ ) dtdu = X v ∈F ( P ) lim λ →∞ Z N ( v ) τ ( P ) Z P (( λP ) t,u ∩ L = ∅ ) dtdu where in the second line we used Lemma 4.1. Because the probability isbounded by 1, Lebesgues dominated convergence theorem can be appliedyieldinglim λ →∞ W ( λP ) − E W (( λP ) L ) = X v ∈F ( P ) Z N ( v ) τ ( P ) Z lim λ →∞ P (( λP ) t,u ∩ L = ∅ ) dtdu.
13y the translation invariance of the measure dL on L , P (( λP ) t,u ∩ L = ∅ ) = P (( λ ( P − v )) t,u ∩ L = ∅ )and the set λ ( P − v ) converges to an infinite cone C v with apex in the origin,as λ → ∞ . Thus for u ∈ N ( v ) and t fixed we havelim λ →∞ P (( λP ) t,u ∩ L = ∅ ) = P ( C v ) t,u ∩ L = ∅ ) , and the mean width difference converges tolim λ →∞ W ( λP ) − E W (( λP ) L ) = X v ∈F ( P ) Z N ( v ) τ ( P ) Z P (( C v ) t,u ∩ L = ∅ ) dtdu = γ ( P ) . We need some argument that the probability P (( C v ) t,u ∩ L = ∅ ) is not van-ishing. By Lemma 3.3, P (( C v ) t,u ∩ L = ∅ ) ≥ − V d (( C v ) t,u ) . Observe that ( C v ) t,u is a pyramid with height t , and thus the volume tends to0 for t →
0. Therefore the probability is bounded from below by a functionwhich is strictly positive in a neighborhood of t = 0, hence c ( P ) must bepositive. References [1] B´ar´any, I.: The chance that a convex body is lattice-point free: A rela-tive of Buffon’s needle problem.
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