K´ aroly Bezdek

Minimizing the mean projections of finite ρ-separable packings ^∗†

K´ aroly Bezdek

and Zsolt L´ angi

December 14, 2017

Abstract

A packing of translates of a convex body in thed-dimensional Euclidean spaceE^dis said to be totally separable if any two packing elements can be separated by a hyperplane ofE^ddisjoint from the interior of every packing element. We call the packingP of translates of a centrally symmetric convex bodyC inE^daρ-separable packing for givenρ≥1 if in every ball concentric to a packing element ofP having radiusρ(measured in the norm generated byC) the corresponding sub-packing ofPis totally separable.

The main result of this paper is the following theorem. Consider the convex hullQofnnon-overlapping translates of an arbitrary centrally symmetric convex bodyCforming aρ-separable packing inE^dwith n being sufficiently large for givenρ≥1. If Qhas minimal mean i-dimensional projection for given i with 1≤i < d, thenQis approximately ad-dimensional ball. This extends a theorem of K. B¨or¨oczky Jr. [Monatsh. Math. 118(1994), 41–54] from translative packings to ρ-separable translative packings forρ≥1.

1 Introduction

We denote thed-dimensional Euclidean space byE^d. LetB^d denote the unit ball centered at the origino in E^d. Ad-dimensional convex bodyC is a compact convex subset ofE^d with non-empty interior intC. (If d= 2, thenCis said to be aconvex domain.) IfC=−C, where−C={−x:x∈C}, thenCis said to be o-symmetricand a translatec+C ofCis called centrally symmetric with centerc.

The starting point as well as the main motivation for writing this paper is the following elegant theorem of B¨or¨oczky Jr. [8]: Consider the convex hull Q of n non-overlapping translates of an arbitrary convex bodyC in E^d with n being sufficiently large. IfQ has minimal mean i-dimensional projection for given i with 1≤i < d, thenQis approximately ad-dimensional ball. In this paper, our main goal is to prove an extension of this theorem to ρ-separable translative packings of convex bodies inE^d. Next, we define the concept ofρ-separable translative packings and then state our main result.

A packing of translates of a convex domain C in E² is said to betotally separable if any two packing elements can be separated by a line ofE²disjoint from the interior of every packing element. This notion was introduced by G. Fejes T´oth and L. Fejes T´oth [9] . We can define a totally separable packing of translates of ad-dimensional convex bodyC in a similar way by requiring any two packing elements to be separated by a hyperplane inE^d disjoint from the interior of every packing element [6, 7].

Definition 1. LetCbe ano-symmetric convex body ofE^d. Furthermore, letk·kCdenote the norm generated byC, i.e., letkxkC:= inf{λ|x∈λC}for any x∈E^d. Now, letρ≥1. We say that the packing

P_sep:={c_i+C |i∈I withkc_j−c_kk_C≥2 for allj6=k∈I}

∗Keywords: totally separable packing, translative packing, density,ρ-separable packing, convex body, volume, mean projec- tion.

†2010 Mathematics Subject Classification: 52C17, 05B40, 11H31, and 52C45.

‡Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.

§Partially supported by the National Research, Development and Innovation Office, NKFI, K-119670.

of (finitely or infinitely many) non-overlapping translates of C with centers {ci | i ∈ I} is a ρ-separable packinginE^d if for eachi∈I the finite packing {cj+C |cj+C⊆ci+ρC} is a totally separable packing (inc_i+ρC). Finally, letδ_sep(ρ,C) denote the largest density of allρ-separable translative packings ofC in E^d, i.e., let

δsep(ρ,C) := sup

Psep

lim sup

λ→+∞

P

ci+C⊂W^d_λvold(ci+C) vold(W_λ^d)

! ,

whereW^d_λ denotes the d-dimensional cube of edge length 2λcentered at oinE^d having edges parallel to the coordinate axes ofE^d andvold(·) refers to thed-dimensional volume of the corresponding set inE^d. Remark 1. Letδ(C)(resp., δsep(C)) denote the supremum of the upper densities of all translative packings (resp., totally separable translative packings) of the o-symmetric convex body C in E^d. Clearly, δsep(C)≤ δsep(ρ,C)≤δ(C)for allρ≥1. Furthermore, if1≤ρ <3, then any ρ-separable translative packing of C in E^d is simply a translative packing of C and therefore,δsep(ρ,C) =δ(C).

Recall that the meani-dimensional projectionM_i(C) (i= 1,2, . . . , d−1) of the convex bodyC in E^d, can be expressed ([13]) with the help of mixed volume via the formula

Mi(C) = κi

κ_dV(

i

z }| { C, . . . ,C,

d−i

z }| { B^d, . . . ,B^d),

where κ_d is the volume of B^d in E^d. Note that M_i(B^d) = κ_i, and the surface area of C is S(C) =

dκ_d

κ_d−1M_d−1(C) and in particular, S(B^d) = dκ_d. Set M_d(C) := vol_d(C). Finally, let R(C) (resp., r(C)) denote the circumradius (resp., inradius) of the convex bodyC in E^d, which is the radius of the smallest (resp., largest) ball that contains (resp., is contained in)C. Our main result is the following.

Theorem 1. Let d ≥ 2, 1 ≤ i ≤ d−1, ρ ≥ 1, and let Q be the convex hull of the ρ-separable packing of n translates of theo-symmetric convex body C inE^d such thatMi(Q) is minimal and n≥ _δ ^4dd4d

sep(ρ,C)^d−1 ·

ρ^R(C)_r(C)d

. Then

r(Q)

R(Q) ≥1− ω n^d(d+3)2

, (1)

forω=λ(d)_ρR(C)

r(C)

_d+3²

, whereλ(d)depends only on the dimensiond. In addition,

Mi(Q) =

1 + σ n^1d

Mi(B^d)

vold(C) δsep(ρ,C)κd

_dⁱ

·n^di,

where− 2.25R(C)ρdi

r(C)δ_sep(ρ,C) ≤σ≤_r(C)δ^2.1R(C)ρi

sep(ρ,C).

Remark 2. It is worth restating Theorem 1 as follows: Consider the convex hull Q of n non-overlapping translates of an arbitrary o-symmetric convex body C forming a ρ-separable packing in E^d with n being sufficiently large. If Q has minimal mean i-dimensional projection for given i with 1 ≤ i < d, then Q is approximately a d-dimensional ball.

Remark 3. The nature of the analogue question on minimizingM_d(Q) = vol_d(Q)is very different. Namely, recall that Betke and Henk [4] proved L. Fejes T´oth’s sausage conjecture for d≥42 according to which the smallest volume of the convex hull of n non-overlapping unit balls in E^d is obtained when the n unit balls form a sausage, that is, a linear packing (see also [2] and [3]). As linear packings of unit balls areρ-separable therefore the above theorem of Betke and Henk applies toρ-separable packings of unit balls inE^dfor allρ≥1 and d ≥42. On the other hand, the problem of minimizing the volume of the convex hull of n unit balls forming a ρ-separable packing in E^d remains an interesting open problem for ρ≥1 and 2 ≤d <42. Last but not least, the problem of minimizingMd(Q)foro-symmetric convex bodiesCdifferent from a ball inE^d seems to be wide open forρ≥1 andd≥2.

Remark 4. Let d ≥ 2, 1 ≤ i ≤ d−1, n > 1, and let C be a given o-symmetric convex body in E^d. Furthermore, let Q be the convex hull of the totally separable packing ofn translates of C inE^d such that M_i(Q) is minimal. Then it is natural to ask for the limit shape of Q as n →+∞, that is, to ask for an analogue of Theorem 1 within the family of totally separable translative packings of C in E^d. This would require some new ideas besides the ones used in the following proof of Theorem 1.

In the rest of the paper by adopting the method of B¨or¨oczky Jr. [8] and making the necessary modifica- tions, we give a proof of Theorem 1.

2 Basic properties of finite ρ-separable translative packings

The following statement is theρ-separable analogue of the Lemma in [5] (see also Theorem 3.1 in [2]).

Lemma 1. Let{ci+C| 1≤i≤n} be an arbitraryρ-separable packing ofntranslates of the o-symmetric convex bodyC inE^d withρ≥1,n≥1, andd≥2. Then

nvold(C)

vol_d(∪ⁿ_i=1c_i+ 2ρC)≤δsep(ρ,C).

Proof. We use the method of the proof of the Lemma in [5] (resp., Theorem 3.1 in [2]) with proper modi- fications. The details are as follows. Assume that the claim is not true. Then there is an >0 such that

vol_d(∪ⁿ_i=1c_i+ 2ρC) = nvol_d(C)

δsep(ρ,C)− (2)

LetCn={ci |i= 1, . . . , n}and let Λ be a packing lattice ofCn+ 2ρC=∪ⁿ_i=1ci+ 2ρCsuch thatCn+ 2ρC is contained in a fundamental parallelotope of Λ say, inP, which is symmetric about the origin. Recall that for eachλ >0,W_λ^d denotes thed-dimensional cube of edge length 2λcentered at the originoinE^d having edges parallel to the coordinate axes of E^d. Clearly, there is a constantµ > 0 depending onP only, such that for eachλ >0 there is a subsetL_λof Λ with

W^d_λ⊆Lλ+PandLλ+ 2P⊆W^d_λ+µ . (3) The definition ofδ_sep(ρ,C) implies that for eachλ >0 there exists aρ-separable packing ofm(λ) translates ofCin E^d with centers at the points ofC(λ) such that

C(λ) +C⊂W_λ^d

and

lim

λ→+∞

m(λ)vold(C)

vold(W^d_λ) =δ_sep(ρ,C). As lim_λ→+∞^vold(W

d λ+µ)

vol_d(W^d_λ) = 1 therefore there existξ >0 and aρ-separable packing ofm(ξ) translates ofCin E^d with centers at the points ofC(ξ) and withC(ξ) +C⊂W^d_ξ such that

vol_d(P)δ_sep(ρ,C)

vold(P) + < m(ξ)vol_d(C)

vold(W_ξ+µ^d ) and nvol_d(C)

vold(P) + < nvol_d(C)card(L_ξ)

vold(W^d_ξ+µ) , (4) where card(·) refers to the cardinality of the given set. Now, for eachx∈Pwe define aρ-separable packing ofm(x) translates ofC inE^d with centers at the points of

C(x) :={x+L_ξ+C_n} ∪ {y∈C(ξ)|y∈/ x+L_ξ+C_n+ int(2ρC)} . Clearly, (3) implies that C(x) +C ⊂ W^d_ξ+µ. Now, in order to evaluate R

x∈Pm(x)dx, we introduce the functionχy for eachy∈C(ξ) defined as follows: χy(x) = 1 if y∈/ x+Lξ+Cn+ int(2ρC) and χy(x) = 0

for any other x ∈ P. Based on the origin symmetric P it is easy to see that for any y ∈ C(ξ) one has R

x∈Pχy(x)dx= vold(P)−vold(Cn+ 2ρC). Thus, it follows in a straightforward way that Z

x∈P

m(x)dx= Z

x∈P

ncard(L_ξ) + X

y∈C(ξ)

χ_y(x)

dx=nvol_d(P)card(L_ξ) +m(ξ) vol_d(P)−vol_d(C_n+ 2ρC) .

Hence, there is a pointp∈Pwith m(p)≥m(ξ)

1−vold(Cn+ 2ρC) vold(P)

+ncard(Lξ) and so

m(p)vold(C)

vold(W^d_ξ+µ) ≥m(ξ)vold(C) vold(W^d_ξ+µ)

1−vold(Cn+ 2ρC) vold(P)

+nvold(C)card(Lξ)

vold(W^d_ξ+µ) . (5) Now, (2) implies in a straightforward way that

vold(P)δsep(ρ,C) vold(P) +

1−vold(Cn+ 2ρC) vold(P)

+ nvold(C)

vold(P) + =δsep(ρ,C) (6) Thus, (4), (5), and (6) yield that

m(p)vold(C)

vold(W^d_ξ+µ) > δsep(ρ,C).

AsC(p) +C⊂W_ξ+µ^d this contradicts the definition of δsep(ρ,C), finishing the proof of Lemma 1.

Definition 2. Let d ≥ 2, ρ ≥ 1, and let K (resp., C) be a convex body (resp., an o-symmetric convex body) inE^d. Then letνC(ρ,K)denote the largestnwith the property that there exists aρ-separable packing {ci+C | 1≤i≤n} such that{ci |1≤i≤n} ⊂K.

Lemma 2. Let d≥2,ρ≥1, and let K(resp., C) be a convex body (resp., ano-symmetric convex body) in E^d. Then

1 + 2ρR(C) r(K)

−d

vol_d(C)ν_C(ρ,K)

δsep(ρ,C) ≤vol_d(K)≤ vol_d(C)ν_C(ρ,K) δsep(ρ,C) . Proof. Observe that Lemma 1 and the containments K+2ρC ⊆

1 + ^2ρR(C)_r(K)

K yield the lower bound immediately.

We prove the upper bound. Let 0 < ε < δsep(ρ,C). By the definition of δsep(ρ,C), if λis sufficiently large, then there is aρ-separable packing{ci+C|1≤i≤n}such thatCn:={ci |1≤i≤n} ⊂W_λ^d and

nvold(C)

vold(W^d_λ) ≥δsep(ρ,C)−ε. (7)

Sublemma 1. If XandY are convex bodies inE^d andC is an o-symmetric convex body in E^d, then ν_C(ρ,Y)≥ vold(Y)νC(ρ,X)

vol_d(X−Y) . (8)

Proof. Indeed, consider any finite point setX :={x1, . . . ,xN} ⊂X. Observe that the following are equiva- lent for a positive integerk:

• k is the maximum number a point ofX−Y is covered by the setsxi−Y,xi∈X,

• k is the maximum number such that card((z+Y)∩X) =k for some pointz∈X−Y.

Thus,Nvold(Y)≤card((z+Y)∩X) vold(X−Y) for some z∈X−Y. Hence, if{xi+C|1≤i≤N} is an arbitraryρ-separable packing withX={x1, . . . ,xN} ⊂X, then

ν_C(ρ,Y)≥card((z+Y)∩X)≥ vol_d(Y)N vold(X−Y), which implies (8).

Applying (8) to X=W^d_λ andY=−Kand using (7), we obtain νC(ρ,K)≥ nvold(K)

vold(W^d_λ+K) ≥ vold(K)

vold(W_λ+R(K)^d )·vold(W^d_λ)(δsep(ρ,C)−ε) vold(C) , which finishes the proof of Lemma 2.

Definition 3. Let d≥2,n≥1,ρ≥1, and letC be an o-symmetric convex body in E^d. Then letR_C(ρ, n) be the smallest radius R >0 with the property thatνC(ρ, RB^d)≥n.

Clearly, for anyε >0 we haveνC(ρ,(RC(ρ, n)−ε)B^d)< n, and thus, by Lemma 2 (forK=RC(ρ, n)B^d), we obtain

Corollary 1. Letd≥2,n≥1,ρ≥1, and letC be ano-symmetric convex body in E^d. Then R_C(ρ, n)^d≤ vol_d(C)n

δsep(ρ,C)κd

≤(R_C(ρ, n) + 2ρR(C))^d. (9)

Lemma 3. Let n≥^{4dδsep(ρ,C)ρ}_r(C)d^dR(C)d andi= 1,2, . . . , d−1. Then forR=RC(ρ, n),

Mi((R+ρR(C))B^d)≤Mi(B^d)

vold(C)n δ_sep(ρ,C)κ_d

_dⁱ

1 + 2δsep(ρ,C)^1dρR(C)

r(C) · 1

n^1d

!i

.

Proof. Sett=R+ 2ρR(C). Then the first inequality in (9) yields that R+ρR(C)≤ t−ρR(C)

t−2ρR(C)

vol_d(C)n δsep(ρ,C)κd

¹_d .

Thus, by the second inequality in (9) and the condition that n≥ ^{4dδsep(ρ,C)ρ}_r(C)d^dR(C)d ≥ ^{4dδsep(ρ,C)ρ}_vol ^dR(C)dκd

d(C) ,

we obtain that t−ρR(C) t−2ρR(C) = 1 +

t ρR(C)−2

−1

≤1 + 2δ_sep(ρ,C)^d1ρR(C)κ_d^1d vold(C)^d1 · 1

n^d1 ≤1 + 2δ_sep(ρ,C)^d1ρR(C)

r(C) · 1

n^1d.

3 Proof of Theorem 1

In the proof that follows we are going to use the following special case of the Alexandrov-Fenchel inequality ([13]): ifK is a convex body inE^d satisfyingM_i(K)≤M_i(rB^d) for given 1≤i < dandr >0, then

Mj(K)≤Mj(rB^d) (10)

holds for allj withi < j ≤d. In particular, this statement for j =dcan be restated as follows: if K is a convex body in E^d satisfyingMd(K) =Md(rB^d) for givend≥2 andr >0, thenMi(K)≥Mi(rB^d) holds for alliwith 1≤i < d.

Letd≥2, 1≤i≤d−1,ρ≥1, and letQbe the convex hull of the ρ-separable packing ofntranslates of theo-symmetric convex bodyCin E^d such thatMi(Q) is minimal and

n≥ 4^dd^4d δsep(ρ,C)^d−1 ·

ρR(C)

r(C) ^d

. (11)

By the minimality ofMi(Q) we have that

Mi(Q)≤Mi(RB^d+C)≤Mi((R+ρR(C))B^d) (12) withR=RC(ρ, n). Note that (12) and Lemma 3 imply that

Mi(Q)≤ 1 + 2δsep(ρ,C)^1dρR(C)

r(C) · 1

n^1d

!i

Mi(B^d)

vold(C) δ_sep(ρ,C)κ_d

_dⁱ

·n^di.

We examine the functionx7→(1 +x)ⁱ, where, by (11), we havex≤x0=_2d¹4. The convexity of this function implies that (1 +x)ⁱ≤1 +i(1 +x0)ⁱ⁻¹x. Thus, from the inequality 1 +_2d¹4

d−1

≤ ³³₃₂<1.05, whered≥2, the upper bound forMi(Q) in Theorem 1 follows.

On the other hand, in order to prove the lower bound for M_i(Q) in Theorem 1, we start with the observation that (10) (based on (12)), (11), and Lemma 3 yield that

S(Q)≤S((R+ρR(C))B^d)≤dκ_d

nvold(C) δ_sep(ρ,C)κ_d

^d−1_d

1 + 2δsep(ρ,C)^1dρR(C)

r(C) · 1

n^1d

!d−1

. (13) Thus, (13) together with the inequalitiesS(Q)r(Q)≥vol_d(Q) (cf. [11]) and vol_d(Q)≥nvol_d(C) yield

r(Q)≥ 1 +2δsep(ρ,C)^1dρR(C)

r(C) · 1

n^1d

!−(d−1)

vold(C)^d1δsep(ρ,C)^d−1d dκ

1 d

d

·n^d1. (14)

Applying the assumption (11) and vol_d(C)≥κ_dr(C)^d to (14), we get that r(Q)≥

1 + 1

2d⁴

−(d−1)

δsep(ρ,C)^d−1d r(C)

d n^1d ≥ 4d³

(1 +_2d¹₄)^d−1R(C)≥31R(C). (15) LetPdenote the convex hull of the centers of the translates ofC inQ. Then, (15) implies

r(P)≥r(Q)−R(C)≥ 30

31r(Q)≥ 8δ_sep(ρ,C)^d−1d r(C)

9d ·n^1d. (16)

Hence, by (16) and Lemma 2,

vold(Q)≥vold(P)≥ 1 + 9dρR(C)

4δ_sep(ρ,C)^d−1d r(C)· 1 n^1d

!−d

· nvold(C)

δ_sep(ρ,C), (17) which implies in a straightforward way that

vold(Q)≥

1 + 9dρR(C) 4δ_sep(ρ,C)r(C)· 1

n^{1d −d}

· nvold(C)

δ_sep(ρ,C). (18)

Note that (10) (see the restated version for j = d) implies that M_i(Q)≥ _vol

d(Q) κd

_dⁱ

κ_i. Then, replacing vold(Q) by the right-hand side of (18), and using the convexity of the functionx7→(1 +x)⁻ⁱ forx > −1 yields the lower bound forMi(Q)in Theorem 1.

Finally, we prove the statement about the spherical shape ofQ, that is, the inequality (1). As in [8], let

θ(d) = 1

2^d+32 √ 2π√

d(d−1)(d+ 3)min

( 3

π²d(d+ 2)2^d, 16 (dπ)^d−14

) .

Using the inequality ^κ_κ^d−1

d ≥q

d

2π (cf. [1]) and (6) of [10], we obtain S(Q)

S(B^d) d

vold(B^d) vold(Q)

!d−1

−1≥θ(d)·

1− r(Q) R(Q)

^d+3₂

(see also (5) of [8]). Substituting (13) and (17) into this inequality, we obtain

1 +2δ_sep(ρ,C)^1dρR(C)

r(C) · 1

n^d1

!d(d−1)

1 + 9dρR(C)

4δsep(ρ,C)^d−1d r(C)· 1 n^1d

!d(d−1)

≥

S(Q) S(B^d)

d

vol_d(B^d) vold(Q)

!d−1

.

By the assumptionsd≥2 and (11), it follows that 4d²(d−1) ρR(C)

δsep(ρ,C)r(C)· 1

n^1d ≥θ(d)

1− r(Q) R(Q)

^d+3₂

. (19)

Note that by [12], _δ ¹

sep(ρ,C) ≤

2^3d2· v u u u t



 d(d+1)

2

d





(d+1)^d2π^d2Γ(^d₂⁺¹) . This and (19) implies (1), finishing the proof of Theorem 1.

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K´aroly Bezdek

Department of Mathematics and Statistics, University of Calgary, Calgary, Canada Department of Mathematics, University of Pannonia, Veszpr´em, Hungary

bezdek@math.ucalgary.ca

and Zsolt L´angi

MTA-BME Morphodynamics Research Group and Department of Geometry Budapest University of Technology and Economics, Budapest, Hungary zlangi@math.bme.hu