**A LOCAL MINIMUM ENERGY CONDITION OF HEXAGONAL CIRCLE**
**PACKING**

KANYA ISHIZAKA FUJIXEROXCO., LTD.

430 SAKAI, NAKAI-MACHI, ASHIGARAKAMI-GUN

KANAGAWA259-0157, JAPAN. Kanya.Ishizaka@fujixerox.co.jp

*Received 21 September, 2007; accepted 10 July, 2008*
*Communicated by S.S. Dragomir*

ABSTRACT. A sufficient condition for the energy of a point such that a local minimum of the energy exists at every triangular lattice point is obtained. The condition is expressed as a certain type of strong convexity condition of the function which defines the energy. New results related to Riemann sum of a function with such the convexity and new inequalities related to sums on triangular lattice points are also presented.

*Key words and phrases: Packing, Energy, Convex function.*

*2000 Mathematics Subject Classification. 26D15, 74G65.*

**1. I****NTRODUCTION**

In some scientific or engineering fields, we are sometimes required to give or measure well- distributed objects in a space. From a purely mathematical point of view, these requirements are satisfied by solving a question which asks whether the well-distributed points are given by the minimization of a total energy of arbitrarily distributed points. In [7], assuming the well- distributed points to be arranged as in a periodic sphere packing [10, pp.25], we have obtained the minimum energy condition in a one-dimensional case; this condition is given as a certain strong convexity condition of the function which defines the energy. A natural question arising in this context is whether the one-dimensional condition can be theoretically extended to higher dimensional spaces.

In this study, we consider the two-dimensional case by imposing two strong restrictions. The first constraint restricts the packing structure to a hexagonal circle packing. Although general circle packing structures are unknown [5, D1], the densest (ideal) circle packing is achieved by the hexagonal circle packing [10] [11, Theorem 1.3 (Lagrange (1773), Thue (1910), L. Fejes Tóth (1940), Segre and Mahler (1944))], which is equivalent to the structure with the center of each circle placed on the triangular lattice points. The second constraint restricts the minimum energy analyses to the point-based local minimum analysis, which addresses whether a small

The author would like to thank the referee for helpful suggestions, especially the suggestions on the simplification of the proof of Lemma 4.1.

292-07

perturbation of a point increases the energy of the point. These restrictions are motivated by a suggestion about the study of local minima for optimal configurations [5, F17].

Hence, we investigate the condition for the energy of a point such that each triangular lattice point has a locally optimal configuration with respect to the energy.

**2. D****EFINITION**

**Definition 2.1. For a point set**X ⊂R^{2} andf : (0,∞)→R*, let the energy of a point*x∈Xbe
J(X, x, f) = X

y∈X\{x}

f(kx−yk), wherek · kis the Euclidean norm.

For ease of analysis, we use the above-mentioned definition for defining the energy that is different from the energy in the one-dimensional case [7]. However, the obtained results in this study are also valid for energies having the same form as that of the energy in the one- dimensional case whenXis a finite set andf(0)is defined.

**Definition 2.2. Let**d > 0,v_{1} =

1 2,

√ 3 2

, andv_{2} =

1 2,−

√ 3 2

*. Let one-sixth of the triangular*
*lattice points be given by*

(2.1) Λ_{d}=

d(iv_{1}+jv_{2}) :i∈N, j = 0, . . . , i−1 .

*Let one-sixth of equally spaced points on equally spaced concentric circles be given by*
(2.2) Ωd=

(idcosτij, idsinτij) :τij =π/3·(1−j/i), i∈N, j = 0, . . . , i−1 .
*In addition, let the triangular lattice points* Λ^{∗}_{d} *and equally spaced points on equally spaced*
*concentric circles* Ω^{∗}_{d} be the unions of the rotations of Λ_{d} and Ω_{d}, respectively, around the
origin by angles ^{π}_{3}j forj = 0, . . . ,5.

Figure 2.1(a) and (b) illustratesΛ_{d}andΩ_{d}, respectively. From the definition ofΛ^{∗}_{d}, it can be
easily checked thatΛ^{∗}_{d}={d(iv_{1}+jv_{2}) :i, j ∈Z} \ {0}.

*d*
*dv*1

*dv*2

(a)

*d*

(b)

*Figure 2.1: Illustration of two point sets along with related parameters: (a)*Λ_{d}*and (b)*Ω_{d}*.*

**3. A****NALYTICAL****C****ONDITION FOR****L****OCAL****M****INIMUM** **E****NERGY**

In this section, we derive the analytical condition for the energy J such that it has a local minimum atvwhenXconsists of equally spaced points on each of the concentric circles with arbitrary radii centered atv.

* Proposition 3.1. Let* n ∈ N

*. For*i = 1, . . . , n, let k

_{i}∈ N

*with*k

_{i}≥ 3, θ

_{i}∈ [0,2π), and 0 < r

_{i}≤ 1. For i = 1, . . . , n

*and*j = 0, . . . , k

_{i}−1, let τ

_{ij}= 2πj/k

_{i}+θ

_{i}

*and vectors*u

_{ij}= (r

_{i}cosτ

_{ij}, r

_{i}sinτ

_{ij}). Letv∈R

^{2}

*and a point set*X ⊂R

^{2}

*satisfying*

(3.1) X∩ {y∈R^{2} :|y| ≤1}=

n

[

i=1

{u_{ij} :j = 0, . . . , k_{i}−1}.

*Let*f : (0,∞)→R*belong to the class*C^{2}*and*f(x) = 0*on*x≥1. If

(3.2) X

y∈X

f^{00}(|y|) + f^{0}(|y|)

|y|

=

n

X

i=1

k_{i}

f^{00}(r_{i}) + f^{0}(r_{i})
ri

>0,
*then the energy*J((v+X)∪ {x},x, f)*has a local minimum at*x=v.

*Proof. We analyze the derivative of*J and the Hessian matrix of the derivative. Without loss
of generality, we may assume v = 0 andθ_{i} = 0for eachi because these restrictions do not
influence the value ofJ. Then, the energy of a pointxis given by

J(X∪ {x},x, f) = X

y∈X

f(|x−y|) =

n

X

i=1 ki−1

X

j=0

f(|x−u_{ij}|).

From the assumption,f =f^{0} =f^{00}= 0onx≥1. Thus,J is certainly twice differentiable with
respect tox.

First, we consider∇J. Since the derivative of|x|with respect toxis _{|x|}^{x}, we get

∇J =

n

X

i=1 ki−1

X

j=0

f^{0}(|x−u_{ij}|)· x−u_{ij}

|x−u_{ij}|.

Note that at the pointx =0, we have|x−u_{ij}| =|u_{ij}|= r_{i}. Here, form, p ∈ Nwithm < p
and forη∈Rwithcosη 6= 1, a general exponential sum formula holds inC:

p−1

X

m=0

(cosmη+isinmη) =

p−1

X

m=0

e^{imη}
(3.3)

= 1−e^{ipη}

1−e^{iη} = 1−(cospη+isinpη)
1−(cosη+isinη) .

(In (3.3),idenotes the imaginary unit.) Substitutingm=j, p=k_{i}, andη = 2π/k_{i}in (3.3), we
obtainPki−1

j=0 u_{ij} =0for eachi. Hence,∇J =0holds atx=0. Thus,0is a stationary point.

Next, we analyze the Hessian matrix of∇J to determine whetherJ has a local minimum at
x=0. Using the notationsx= (x_{1}, x_{2})andu_{ij} = (u_{ij1}, u_{ij2}), we get

∂^{2}J

∂x_{m}^{2} =

n

X

i=1 ki−1

X

j=0

f^{00}(|x−u_{ij}|)− f^{0}(|x−u_{ij}|)

|x−u_{ij}|

· (x_{m}−u_{ijm})^{2}

|x−u_{ij}|^{2} +

n

X

i=1 ki−1

X

j=0

f^{0}(|x−u_{ij}|)

|x−u_{ij}| ,
wherem = 1,2and

∂^{2}J

∂x_{2}∂x_{1} = ∂^{2}J

∂x_{1}∂x_{2} =

n

X

i=1 ki−1

X

j=0

f^{00}(|x−u_{ij}|)− f^{0}(|x−u_{ij}|)

|x−u_{ij}|

· (x_{1}−u_{ij1})(x_{2}−u_{ij2})

|x−u_{ij}|^{2} .

Note thatcosη 6= 1from the assumption k_{i} ≥ 3. Hence, by substitutingm = j, p = k_{i}, and
η= 4π/k_{i} in (3.3), we obtain

ki−1

X

j=0

cos 2τ_{ij} =

ki−1

X

j=0

sin 2τ_{ij} = 0

for eachi. Hence, using double-angle formulas, fori= 1, . . . , n, we have

ki−1

X

j=0

u_{ij1}^{2} =

ki−1

X

j=0

u_{ij2}^{2} = k_{i}r_{i}^{2}
2 ,

ki−1

X

j=0

u_{ij1}u_{ij2} = 0.

From these equalities, atx=0, we have _{∂x}^{∂}^{2}^{J}

12 = _{∂x}^{∂}^{2}^{J}

22, _{∂x}^{∂}^{2}^{J}

1∂x2 = _{∂x}^{∂}^{2}^{J}

2∂x1 = 0, and

∂^{2}J

∂x_{1}^{2} =

n

X

i=1 ki−1

X

j=0

f^{00}(r_{i})− f^{0}(r_{i})
r_{i}

·u_{ij1}^{2}
r_{i}^{2} +

n

X

i=1 ki−1

X

j=0

f^{0}(r_{i})
r_{i}

=

n

X

i=1

k_{i}
2

f^{00}(r_{i}) + f^{0}(r_{i})
r_{i}

.
Hence, at x = 0, both the discriminant and the term _{∂x}^{∂}^{2}^{J}

12 are positive from the assumption.

Thus,J has a local minimum atx=0.

We can apply Proposition 3.1 toΛ^{∗}_{d} and Ω^{∗}_{d} because each set can satisfy the form (3.1) for
fixed ki = 6. Furthermore, we can use Λd and Ωd for the estimations of (3.2) because the
values of (3.2) forX = Λ^{∗}_{d}andX = Ω^{∗}_{d}are 6 times those obtained forX = Λ_{d}andX = Ω_{d},
respectively. In particular, substitutingr =d^{−1} in (2.2), onΩ_{d}, we obtain

X

y∈Ω_{d}

f^{00}(|y|) + f^{0}(|y|)

|y|

=

brc

X

i=1

i

f^{00}
i

r

+ r
if^{0}

i r

(3.4)

=r

brc

X

i=1

f^{0}

i r

+ i

rf^{00}
i

r

.

Thus, the local minimum energy condition onΩ_{d}is simplified into the positivity of the sum of a
single-variable function. Since it might be difficult to directly analyze (3.2) with respect toΛ_{d},
we would first analyze the right-hand side of (3.4) forΩ_{d}.

**4. R****IEMANN** **S****UM OF A****F****UNCTION WITH A****C****ERTAIN** **S****TRONG****C****ONVEXITY**

In [7], for the minimum energy analysis in a one-dimensional case, we have proved a variant of a result obtained by Bennett and Jameson [1]. Here, in order to investigate a sufficient con- dition such that the expression (3.4) may be greater than0, we again adopt the same approach.

For a functionf on(0,1], letS_{n}(f)be the upper Riemann sum for the integralR1

0 f resulting from division of[0,1]intonequal subintervals:

S_{n}(f) = 1
n

n

X

i=1

f i

n

.

Theorem 3A in [1] states that iff is increasing and either convex or concave, thenS_{n}(f)de-
creases withn. The same theorem has been independently proved by Kuang [9]. Further related
results have been presented in [1, 3]. Here, we show thatS_{n}(f)also decreases iffis increasing,

f^{0} x^{1}^{2}
x^{1}^{2}2

is convex, andlim_{x→1}f^{0}(x) = 0.

Before presenting the result, we prove the following lemma.

* Lemma 4.1. Let* a < b

*be real numbers and*f : [a, b] → R

*. Let*p ≥ 1

*be a real number. If*f ≥0,f

*is decreasing, and*f(x)

^{p}

*is convex, then*

(4.1) 1

b−a Z b

a

f(x)dx≤ p

p+ 1f(a) + 1

p+ 1f(b).

*Equality holds iff any one of the following conditions is satisfied:*

(a) p= 1*and*f *is linear on*[a, b];

(b) f *is constant on*[a, b]; and

(c) f(x)^{p} *is linear on*[a, b]*and*f(b) = 0.

*Proof. We show that in fact a stronger inequality*

(4.2) f(xa+ (1−x)b)≤x^{1}^{p}f(a) +

1−x^{1}^{p}
f(b)

holds, where0≤x≤1. By integrating (4.2) overx∈[0,1], we can obtain (4.1).

Ifp= 1, then the result follows from the convexity off.

We assume thatp >1. By using the substitutiong(x) =f(xa+ (1−x)b), it is sufficient to show

(4.3) g(x)≤

1−x^{1}^{p}

g(0) +x^{1}^{p}g(1)
forg : [0,1]→R, whereg ≥0is increasing andg(x)^{p} is convex.

First, consider the case wheng(0) = 0. Sinceg(x)^{p} is convex,g(x)^{p} ≤xg(1)^{p}, thus,

(4.4) g(x)≤x^{p}^{1}g(1)

on[0,1]. Equality holds iffg(x)^{p} is linear; this case corresponds to case (c).

Next, suppose thatg(0) =c >0. If we can show that[g(x)−c]^{p}is convex, then (4.3) follows
from substitutingg(x)−cforg(x)in (4.4). Leth(x) =g(x)^{p} andk(x) = [g(x)−c]^{p}. Since a
convex function is differentiable at all but at most countably many points, we may rely on the
differentiability ofh, and thereforegandk. Then,h^{0}(x) = pg(x)^{p−1}g^{0}(x)and

k^{0}(x) = p[g(x)−c]^{p−1}g^{0}(x) =h^{0}(x)

1− c g(x)

^{p−1}
.

Both h^{0}(x) and (1−c/g(x))^{p−1} are positive and increasing. Hence, k^{0}(x) is increasing, as
required. Equality in (4.3) holds iff

g(x)^{p} = [(1−x^{1/p})g(0) +x^{1/p}g(1)]^{p},
which gives

[g(x)^{p}]^{0} = (g(1)−g(0))

g(1)−g(0) +x^{−1/p}g(0)^{p−1}
.

Here, it follows thatg(0) =g(1)because[g(x)^{p}]^{0}cannot be increasing forp > 1ifg(0) < g(1).

This equality condition corresponds to the condition in case (b).

* Theorem 4.2. Let*f : (0,1]→R

*be differentiable. If*f

*is increasing,*f

^{0}x

^{1}

^{2}x

^{1}

^{2}2

*is convex,*
*and*limx→1f^{0}(x) = 0, thenSn(f)*decreases with*n.

*Proof. From the assumption,* f^{0} ≥ 0 holds and f^{0} x^{1}^{2}

/x^{1}^{2} is decreasing. Without loss of
generality, we may assume thatf(1) = 0and extendf = f^{0} = 0onx≥ 1. For a real number
r≥1, let

S_{r}(f) = 1
r

brc

X

i=1

f i

r

.

We show that S_{r}^{0}(f) ≤ 0forr ≥ 1, whereS_{r}^{0}(f)is the differential coefficient of S_{r}(f)with
respect to r. The existence of S_{r}^{0}(f) is confirmed by the differentiability off on (0,∞)and
f =f^{0} = 0onx≥1. In fact, we have

Sr0

(f) =−1
r^{2}

brc

X

i=1

f

i r

+ i

rf^{0}
i

r

.
The substitutionx=t^{1}^{2} gives

Z b^{1}^{2}
a^{1}^{2}

f^{0}(x)dx=
Z b

a

f^{0} t^{1}^{2}
2t^{1}^{2} dt.

Thus, by applyingf^{0} x^{1}^{2}

x^{1}^{2} tof in Lemma 4.1 withp= 2, for0< a < b, we obtain

(4.5) 1

b−a
Z b^{1}^{2}

a^{1}^{2}

f^{0}(x)dx≤ 2

6 · f^{0} a^{1}^{2}
a^{1}^{2} +1

6 ·f^{0} b^{1}^{2}
b^{1}^{2} .
Substitutinga= ^{j}_{r}2

andb = ^{j+1}_{r} 2

in (4.5), we get f

j+ 1 r

−f j

r

≤ 2

6r · 2j + 1
j f^{0}

j r

+ 1

6r · 2(j+ 1)−1
j+ 1 f^{0}

j+ 1 r

. Summing overj =i, . . . ,brcand usingf(1) = 0, we obtain

(4.6) −f

i r

≤ 1 r

brc

X

j=i

f^{0}
j

r

+ 1 6r

brc

X

j=i

1
jf^{0}

j r

− 1

6r · 2i−1
i f^{0}

i r

.
Thus, from (4.6) andf^{0} ≥0, we obtain

brc

X

i=1

f

i r

+ i

rf^{0}
i

r

=

brc

X

i=1

f i

r

+1 r

brc

X

j=i

f^{0}
j

r

(4.7)

≥ 1 6r

brc

X

i=1

2i−1
i f^{0}

i r

− 1 6r

brc

X

i=1 brc

X

j=i

1
jf^{0}

j r

= 1 6r

brc

X

i=1

i−1
i f^{0}

i r

≥0.

Hence,S_{r}^{0}(f)≤0holds. Thus,S_{r}(f)decreases withr≥1.

From Lemma 4.1, equality in (4.5) holds iff eitherf^{0} = 0on[a, b]or f^{0} x^{1}^{2}
x^{1}^{2}2

is linear
on [a, b] withf^{0} b^{1}^{2}

= 0. Thus, from (4.6) and (4.7), S_{r}^{0}(f) = 0holds iff eitherf^{0} = 0on
[^{1}_{r},1]or f^{0} x^{1}^{2}

x^{1}^{2}2

is linear on[^{1}_{r},^{2}_{r}]withf^{0}(^{2}_{r}) = 0, indicating thatf^{0} = 0on[^{2}_{r},1]. Here,
the latter condition withf(^{1}_{r})6= 0can hold only for one fixedr. Hence, for anyr_{1} ≥1,S_{r}(f)
strictly decreases withr≥r_{1} ifff^{0} >0on 0,_{r}^{1}

1

.

Therefore,S_{n}(f)decreases withniff is increasing and either
(i) f is convex or concave (from [1, Theorem 3A]), or
(ii) f^{0} x^{1}^{2}

x^{1}^{2}2

is convex andlimx→1f^{0}(x) = 0(from Theorem 4.2).

Here, conditions (i) and (ii) are independent of each other, which can be observed in the following examples. Letp >0.

**Example 4.1. Let**f(x) =−(1−x)^{p}andf^{0}(x) = p(1−x)^{p−1}. Then,f is convex if0< p ≤1
and concave if p ≥ 1. Further, f^{0} x^{1}^{2}

x^{1}^{2}2

is convex and lim_{x→1}f^{0}(x) = 0 if p ≥ 1.5.

Further,f^{0} x^{1}^{2}

x^{1}^{2} is convex ifp≥2.

**Example 4.2. Let** f(x) = −(1−x^{2})^{p} and f^{0}(x) = 2px(1− x^{2})^{p−1}. Then, f is convex if
0 < p ≤ 1 and neither convex nor concave if p > 1. Further, f^{0} x^{1}^{2}

x^{1}^{2}2

is convex and
limx→1f^{0}(x) = 0ifp≥1.5. Further,f^{0} x^{1}^{2}

x^{1}^{2} is convex ifp≥2.

* Theorem 4.3. Let*f : (0,1]→R

*. If*f x

^{1}

^{2}

x^{1}^{2} *is decreasing and convex and*f(1) = 0, then
(4.8)

Z 1

1 n

f(x)dx≤ 1 n

n

X

i=1

f i

n

≤ Z 1

0

f(x)dx.

*Proof. Let*a < b. The Hermite-Hadamard inequality for a convex functionf gives
f

a+b 2

≤ 1 b−a

Z b a

f(x)dx≤ f(a) +f(b)

2 .

By applying the convexity off x^{1}^{2}

x^{1}^{2} to this inequality, we obtain
(b−a)f

a+b 2

^{1}_{2}!

·

a+b 2

−^{1}_{2}

≤ Z b

a

f x^{1}^{2}
x^{1}^{2} dx
(4.9)

≤ b−a 2

"

f a^{1}^{2}

a^{1}^{2} +f b^{1}^{2}
b^{1}^{2}

#
.
Substitutinga= (_{n}^{i})^{2} andb= (^{i+1}_{n} )^{2}on the right-hand inequality in (4.9), we get

Z ^{i+1}_{n}

i n

f(x)dx≤ 1

4n ·2i+ 1 i f

i n

+ 1

4n · 2(i+ 1)−1 i+ 1 f

i+ 1 n

. Summing overi=j, . . . , n−1and usingf ≥0, for eachj = 1, . . . , n−1, we obtain

Z 1

j n

f(x)dx≤ 1 n

n

X

i=j

f i

n

− 1

4n ·2j−1

j f

j n

− 1

4n ·2n−1 n f(1) (4.10)

≤ 1 n

n

X

i=j

f i

n

.

Hence, the left-hand inequality in (4.8) follows from (4.10) whenj = 1. Next, we extendf = 0
onx≥1, which yields the convexity off x^{1}^{2}

x^{1}^{2} on(0,∞). Then, substitutinga = ^{i}^{2}_{n}^{−i}2 and
b= ^{i}^{2}_{n}^{+i}2 on the left-hand inequality in (4.9), we get

2i
n^{2}f

i n

· i

n −1

≤
Z ^{i2+i}

n2

i2−i n2

f x^{1}^{2}
x^{1}^{2} dx.

Summing overi= 1, . . . , n, we obtain the right-hand inequality in (4.8).

In (4.10), equalities hold ifff^{0} x^{1}^{2}

x^{1}^{2} is linear on_{j}

n,1

andf _{n}^{j}

= 0, that is,f = 0on
_{j}

n,1

. Hence, equality on the left-hand inequality in (4.8) holds ifff = 0on_{1}

n,1

. Similarly, equality on the right-hand inequality in (4.8) holds ifff = 0on(0,1].

If f is decreasing on [0,1]and f(1) = 0, then (4.8) and R1

0 f ≤ ^{1}_{n}Pn−1
i=0 f _{n}^{i}

are trivial.

However, when we use a functionf on[0,1]in Theorem 4.3, such an additional upper estima-
tion no longer holds. From (4.10), if a stronger condition thatf^{0} x^{1}^{2}

x^{1}^{2} is convex is assumed
in Theorem 4.2, then in (4.7),

f i

r

+1 r

brc

X

j=i

f^{0}
j

r

≥0 holds for eachi.

**5. L****OCAL****M****INIMUM** **E****NERGY** **C****ONDITION**

We can obtain the local minimum energy condition for Ω^{∗}_{d} from the result obtained in the
previous section.

* Proposition 5.1. Let*0 < d ≤ 1

*and*Ω

^{∗}

_{d}

*be defined by Definition 2.2. Let*v ∈ R

^{2}

*and a point*

*set*X ⊂R

^{2}

*satisfying*

{x∈X :|x| ≤1}={x∈Ω^{∗}_{d}:|x| ≤1}.

*Let*f : (0,∞) → R*belong to the class*C^{2} *with*f = 0*on*x ≥ 1*and*f^{00} 6≡ 0*on*[d,1]. Iff *is*
*convex and either*

(i) f^{0} *is concave or*
(ii) f^{00} x^{1}^{2}

x^{1}^{2}2

*is convex and either is strictly convex on*[d,2d]*or*f(2d)6= 0,
*then the energy*J((v+X)∪ {x},x, f)*has a local minimum at*x=v.

*Proof. Let*r = d^{−1}. As stated after Proposition 3.1, it is sufficient to show that (3.4) is greater
than0. In case (i),f^{00}is decreasing. Moreover, there is an interval contained in[d,1]in which
f^{00}is strictly decreasing becausef^{00} 6≡0on[d,1]andf^{00}(1) = 0. Thus,

brc

X

i=1

f^{0}

i r

+ i

rf^{00}
i

r

=

brc

X

i=1

−

Z 1

i r

f^{00}(x)dx+1
r

brc

X

j=i

f^{00}
j

r

>0.

In case (ii), the result follows from (4.7) and related arguments presented after that.

It is expected that a result similar to that of Proposition 5.1 can be obtained for the triangular
lattice pointsΛ^{∗}_{d}being similar in structure toΩ^{∗}_{d}, thereby leading to the following theorem. In
the proof, two inequalities related to the triangular lattice points are required. The proofs of
these inequalities are given in Section 7. In the statement of Theorem 5.2, a specific value ofp
is given. The meaning of the valuepis explained in the proof of the theorem.

* Theorem 5.2. Let*0< d ≤1

*and*Λ

^{∗}

_{d}

*be defined by Definition 2.2. Let*v∈ R

^{2}

*and a point set*X ⊂ R

^{2}

*satisfying*{x ∈ X : |x| ≤ 1} ={x ∈ Λ

^{∗}

_{d}: |x| ≤ 1}. Letf : (0,∞) → R

*belong to*

*the class*C

^{2}

*with*f = 0

*on*x≥1

*and*f

^{00}6≡0

*on*[d,1]. Iff

*is convex and either*

(i) f^{0} *is concave or*
(ii) f^{00} x^{1}^{2}

x^{1}^{2}p

*is convex for*p= _{11}^{2} (47 + 27√

3) = 17.048. . .,
*then the energy*J((v+X)∪ {x},x, f)*has a local minimum at*x=v.

*Proof. As stated in Section 3, we can use the one-sixth version set*Λdinstead of the triangular
lattice points setΛ^{∗}_{d}. Thus, from Proposition 3.1, it is sufficient to show that

(5.1)

∞

X

i=1

f^{00}(a_{i}) + f^{0}(a_{i})
ai

>0,

where the sequence{a_{i}}is obtained by sorting the value|y|for ally∈Λ_{d}in increasing order.

More precisely, eacha_{i} is defined by
a_{i} = max

|y|:y∈Λ_{d}, #{z∈Λ_{d}:|z|<|y|}< i .
The first 10 values ofa_{i}ared,√

3d,2d,√ 7d,√

7d,3d,√ 12d,√

13d,√

13dand4d; these values are illustrated in Figure 7.1 in Section 7.

First, we summarize the inequalities that are required for the proof. From Theorem 7.1, which will be stated later in Section 7, for alln∈N,

(5.2)

n

X

i=1

a_{n+1}
a_{i} <2n.

In addition, from Theorem 7.3, forn∈Nwithn≥4, (5.3)

n

X

i=1

a_{n+1}^{2}
a_{i} <

n

X

i=1

3a_{i}.

Sincea_{n}increases withn, for alln ∈Nand any real numberp≥0, we have

n

X

i=1

p

p+ 1 · an+12

a_{i} + 1

p+ 1 · an2

a_{i}

≤

n

X

i=1

an+12

a_{i} .
Thus, by substitutingp= _{11}^{2}(47 + 27√

3)and from (5.3), for alln∈N, we obtain (5.4)

n

X

i=1

p

p+ 1 · a_{n+1}^{2}
a_{i} + 1

p+ 1 · a_{n}^{2}
a_{i}

≤

n

X

i=1

3a_{i},

where equality holds iffn = 3. Note that (5.4) strictly holds for any (large)p≥0whenn6= 3.

The specific value ofpis the upper bound ofpfor satisfying (5.4) whenn = 3.

Next, we prove (5.1) for cases of (i) and (ii) by using (5.2) and (5.4), respectively. From the
assumption, suppose thatf =f^{0} =f^{00} = 0onx≥1andf^{00} 6≡0on[a_{1},1].

**Case (i): Let**f^{0}be concave. Then,f^{00} is decreasing. Thus, for0< a < b,
Z b

a

f^{00}(x)dx≤(b−a)f^{00}(a).

Substitutinga=a_{j} andb=a_{j+1}and summing overj =i, i+ 1, . . ., we obtain

(5.5) f^{0}(a_{i})≥ −

∞

X

j=i

(a_{j+1}−a_{j})f^{00}(a_{j}).

Thus, from (5.2) and (5.5) and considering thatf^{00} = 0onx ≥ 1,f^{00} 6≡ 0on[a_{1},1], andf^{00}is
decreasing, we obtain

∞

X

i=1

f^{00}(ai) + f^{0}(a_{i})
a_{i}

≥

∞

X

i=1

"

f^{00}(ai)−

∞

X

j=i

a_{j+1}−a_{j}

a_{i} f^{00}(aj)

#

=

∞

X

i=1

"

2f^{00}(a_{i})−

i

X

j=1

a_{i+1}
aj

(f^{00}(a_{i})−f^{00}(a_{i+1}))

#

>

∞

X

i=1

h

2f^{00}(a_{i})−2i(f^{00}(a_{i})−f^{00}(a_{i+1}))i

= 0.

**Case (ii): Let**p= _{11}^{2}(47 + 27√

3)and f^{00} x^{1}^{2}
x^{1}^{2}p

be convex. Then,f^{00} x^{1}^{2}

x^{1}^{2} is decreas-
ing since f^{00} ≥ 0and f^{00}(1) = 0. Thus, in the same way as in the derivation of (4.5), from
Lemma 4.1, for0< a < b, we obtain

1 b−a

Z b^{1}^{2}
a^{1}^{2}

f^{00}(x)dx≤ p

2(p+ 1) · f^{00} a^{1}^{2}

a^{1}^{2} + 1

2(p+ 1) · f^{00} b^{1}^{2}
b^{1}^{2} .
Substitutinga=a_{j}^{2}andb=a_{j+1}^{2}and summing overj =i, i+ 1, . . ., we have
(5.6) f^{0}(a_{i})≥ − p

2(p+ 1)

∞

X

j=i

a_{j+1}^{2}−a_{j}^{2}
aj

f^{00}(a_{j})− 1
2(p+ 1)

∞

X

j=i

a_{j+1}^{2}−a_{j}^{2}
aj+1

f^{00}(a_{j+1}).

Thus, from (5.4) and (5.6) and considering thatf^{00}= 0onx≥1,f^{00} 6≡0on[a_{1},1], andf^{00}(x)/x
is decreasing, we obtain

2(p+ 1)

∞

X

i=1

f^{00}(a_{i}) + f^{0}(a_{i})
a_{i}

≥

∞

X

i=1

"

2(p+ 1)f^{00}(ai)−p

∞

X

j=i

a_{j+1}^{2}−a_{j}^{2}

a_{i} ·f^{00}(a_{j})
a_{j}

−

∞

X

j=i

a_{j+1}^{2}−a_{j}^{2}

a_{i} · f^{00}(a_{j+1})
a_{j+1}

#

=

∞

X

i=1

"

3(p+ 1)f^{00}(a_{i})

− p

i

X

j=1

a_{i+1}^{2}
a_{j} +

i

X

j=1

a_{i}^{2}
a_{j}

!

·

f^{00}(a_{i})

a_{i} −f^{00}(a_{i+1})
a_{i+1}

#

>3(p+ 1)

∞

X

i=1

"

f^{00}(ai)−

i

X

j=1

aj

!

·

f^{00}(a_{i})

a_{i} − f^{00}(a_{i+1})
a_{i+1}

#

= 0.

Here, the second inequality certainly holds strictly because in (5.4), strict inequality holds for
n6= 3, andf^{00}(a1)/a1−f^{00}(a2)/a2 >0holds fromf 6≡0on[a1,1].

In cases (i) and (ii), the assumption thatf is convex can be omitted because the other condi-
tions yieldf^{00} ≥0. Nevertheless, it is natural to assume this condition in case (ii).

Now, we address the (second) question presented in the introduction.

**Remark 1. Let us consider the relation between Theorem 5.2 and the one-dimensional re-**
sult [7]. The one-dimensional result was as follows. Consider a finite point setX ⊂R/Zwith
the Euclidean distancek · kdefined by

kx−yk= min{|x−y+e|:e=−1,0,1}

and the energy of X defined by the average value of f(kx −yk) for x, y ∈ X, where f :
[0,1/2] → R. If f is convex, then among all m-point sets for fixed m ≥ 1, the energy is
(globally) minimized by an equally spacedm-point set. Additionally, iff is convex,f^{0} x^{1}^{2}

is
concave, and lim_{x→}^{1}

2 f^{0}(x) = 0, then among all m-point sets for 1 ≤ m ≤ n, the energy is
minimized by an equally spacedn-point set.

It is easy to verify that the condition in Theorem 5.2 is stronger than these one-dimensional conditions. Thus, in the two-dimensional case, even for the existence of a local minimum,

the function f should have a stronger convexity than the convexity which is defined by these one-dimensional conditions.

As stated in Section 2, in the two-dimensional case, by defining an affine transformation
g :R^{2} →R^{2}and the periodic spaceg(R^{2}/Z^{2})with the Euclidean distancek · kdefined by

kx−yk= min{|x−y+e_{1}·g(1,0) +e_{2}·g(0,1)|:e_{1}, e_{2} =−1,0,1},
we can also define the energy of a point as

I(X, x, f) = 1

|X|

X

y∈X

f kx−yk ,

whereXis a point set ing(R^{2}/Z^{2})and|X|is the cardinality ofX. Then, Theorem 5.2 is also
valid for the energyIonly if|X|is finite andf(0)is defined.

**6. E****XAMPLES**

**Remark 2. If**p > 0,f(x)≥0, andf(x)^{p} is convex, thenf(x)^{q}is convex for allq≥p. This is
because forg(x) = x^{q/p},g is increasing and convex on[0,∞). Thus,

g(f(ax+by)^{p})≤g(af(x)^{p}+bf(y)^{p})≤ag(f(x)^{p}) +bg(f(y)^{p})
holds fora, b∈[0,1]witha+b = 1.

**Example 6.1. For**n ∈N, letω_{n}(r) =π^{n}^{2}r^{n}

Γ(^{n}_{2} + 1)denote the volume of an n-dimensional
ball of radius r. Let V_{n}(x) be the volume of the cross region of two identical n-dimensional
balls of unit diameter with their centers at distancerfrom each other.

V_{n}(r) = 2
Z ^{1}_{2}

r 2

ωn−1

r1
4 −x^{2}

!

dx= π^{n−1}^{2}
2^{n−1}Γ ^{n+1}_{2}

Z 1 r

(1−x^{2})^{n−1}^{2} dx.

By omitting the constant coefficient of V_{n}(r), we define g_{n}(r) = R1

r(1 − x^{2})^{(n−1)/2}dx for
0≤ r≤ 1and further extendg_{n}(r) = 0forr > 1. Then, eachg_{n}(x)on[0,1]forn = 1, . . . ,5
is given by

g_{1}(x) = 1−x,
g_{2}(x) = 1

2cos^{−1}x− 1

4sin(2 cos^{−1}x) = 1

2cos^{−1}x− 1
2x√

1−x^{2},
g_{3}(x) = 2

3−x+ 1
3x^{3},
g_{4}(x) = 3

8cos^{−1}x− 1

4sin(2 cos^{−1}x) + 1

32sin(4 cos^{−1}x)

= 3

8cos^{−1}x− 1
2x√

1−x^{2}+1

8x(2x^{2} −1)√

1−x^{2},
g_{5}(x) = 8

15−x+ 2

3x^{3}− 1
5x^{5}.

Forp >0, letf_{np}(x) =g_{n}(x)^{p}. Then,f_{np}(x)is convex for alln≥1andp >0.

Table 6.1 shows the conditions required forpto satisfy the convexities in Proposition 5.1 and
Theorem 5.2 with respect tof_{np}^{0}andf_{np}^{00}under the restrictionf_{np} =f_{np}^{0} =f_{np}^{00}= 0onx= 1
forn = 1, . . . ,5. In the table, the values indicated with an asterisk are approximation values
obtained from the numerical analysis, while the others are exact values. In this example, among
cases (i) and (ii) in Proposition 5.1 or Theorem 5.2, we may confirm that case (ii) is more valid
than the case (i) whenn ≥2. In particular, case (ii) is valid for allp ≥1ifn ≥4. In the case

*Table 6.1: Convexities with respect to*fnp0*and*fnp00*(* results obtained from numerical analysis).*

f_{np}^{0}(x) f_{np}^{0}(x^{1}^{2}) (f_{np}^{00}(x^{1}^{2})/x^{1}^{2})^{17.048} (f_{np}^{00}(x^{1}^{2})/x^{1}^{2})^{2} f_{np}^{00}(x^{1}^{2})/x^{1}^{2}
is concave is concave is convex is convex is convex

n= 1 p > 2 p >2 p≥2.06^{∗} p≥2.5 p≥3

n= 2 p≥2.44^{∗} p > ^{4}_{3} p≥1.38^{∗} p≥ ^{5}_{3} p≥2

n= 3 p≥2.57^{∗} p >1 p≥1.03^{∗} p≥1.25 p≥1.5

n= 4 p≥2.64^{∗} p≥1 p≥1 p≥1 p≥1.2

n= 5 p≥2.68^{∗} p≥1 p≥1 p≥1 p≥1

ofn = 3andp = 1, the two-dimensional condition of Theorem 5.2 is not satisfied, while the one-dimensional condition mentioned in Remark 1 is satisfied.

**7. I****NEQUALITIES****R****ELATED TO****S****UMS ON** **T****RIANGULAR****L****ATTICE** **P****OINTS**

In the rest of the paper, we focus on a variation of lattice point problems to prove (5.2) and (5.3). In lattice point theory, the well-known Gauss’ (lattice point or circle) problem is the problem of counting up the number of square lattice points which are inside a circle of radiusr centered at the origin [6, F1] [8]. Meanwhile, the lattice sum is the problem of determining the sums of a variety of quantities on lattice points [2, Chap. 9]. Although it is not clearly defined, the lattice sum usually targets infinite sums. Our problem may occupy an intermediate position between the two problems because we will investigate a relation between certain lattice sums of finite type and the number of triangular lattice points which are inside a circle.

Hereafter, the interval of the lattice is fixed atd = 1 because the inequalities (5.2) and (5.3)
are not influenced byd. These inequalities can be analyzed by an appropriate approximation of
a_{i} onΛ_{1} as follows.

**Remark 3. Let**{a_{i}}be a sequence of the values of|v|forv ∈ Λ_{1} sorted in increasing order.

To obtain an approximation for{a_{i}}, let us consider the case that there arek triangular lattice
points in a circle of radiusr > 1 centered at the origin. Then, the area of the circle,πr^{2}, can
be approximated by the total area ofkidentical equilateral triangles of the area√

3/2. Here, if
r=a_{i}, we havek = 6i. Thus, we havea_{i} ≈b_{i}, where

bi = 3^{3}^{4} ·π^{−}^{1}^{2} ·i^{1}^{2}.
Next, we consider{b_{i}}. Sincex^{−}^{1}^{2} is decreasing,

(7.1) 1

(i+ 1)^{1}^{2} <

Z i+1 i

1

x^{1}^{2}dx < 1
i^{1}^{2}.

Considering thatx^{−}^{1}^{2} is decreasing, and from the left-hand inequality in (7.1), we have
1

i^{1}^{2} <− 1

(i+ 1)^{1}^{2} + 2
i^{1}^{2}
(7.2)

<2(i+ 1)^{1}^{2} −2i^{1}^{2} − 2

(i+ 1)^{1}^{2} + 2
i^{1}^{2}

= 2i

(i+ 1)^{1}^{2} − 2(i−1)
i^{1}^{2} .

Likewise, from the right-hand inequality in (7.1), we have

(7.3) 2(i+ 1)^{1}^{2} − 2(i−1)

i^{1}^{2} < 3
i^{1}^{2}.

Thus, summing each of (7.2) and (7.3) multiplied byioveri= 1, . . . , n, we obtain

n

X

i=1

b_{n+1}

b_{i} <2n <

n

X

i=1

3b_{i}
b_{n+1}.

Hence, if we use the sequence{b_{i}}instead of the sequence{a_{i}}, then (5.2) and (5.3) holds on
the basis of the local inequalities (nearbyi^{1}^{2}) obtained from the concavity ofx^{1}^{2}.

For convenience, we also prepare the representation of the triangular lattice points by means of number theory [4, pp.110]. LetN(n)denote the number of triangular lattice points placed at distance√

nfrom the origin. LetN^{0}(n) = N(n)/6. Then,N^{0}(n)is specified by the following
values:

N^{0}(3^{a}) = 1 for a≥0,

N^{0}(p^{a}) = a+ 1 for p≡1 (mod 3),

N^{0}(p^{a}) = 0 for p≡2 (mod 3), aodd,
N^{0}(p^{a}) = 1 for p≡2 (mod 3), aeven,

wherep6= 3is prime. That is, by factorizing the natural numberninto prime factors by
n= 3^{a}·p_{1}^{b}^{1}· · ·p_{k}^{b}^{k} ·q_{1}^{c}^{1}· · ·q_{l}^{c}^{l},

wherep_{1}, . . . , p_{k} ≡1 (mod 3)andq_{1}, . . . , q_{l}≡2 (mod 3), we have

N^{0}(n) =N^{0}(3^{a})·N^{0}(p_{1}^{b}^{1})· · ·N^{0}(p_{k}^{b}^{k})·N^{0}(q_{1}^{c}^{1})· · ·N^{0}(q_{l}^{c}^{l}).

For example,N^{0}(27) = N^{0}(3^{3}) = 1,N^{0}(39) =N^{0}(3^{1})·N^{0}(13^{1}) = 2, andN^{0}(49) = N^{0}(7^{2}) =
3. Figure 7.1 shows the distances of points inΛ_{1}∪ {0}from the origin fori≤9.

0

1 *√*

3 *√*

7 *√*

13 *√*
21 *√*

31 *√*
43 *√*

57 *√*
73

2 *√*

7 *√*

12 *√*
19 *√*

28 *√*
39 *√*

52 *√*
67

3 *√*

13 *√*
19 *√*

27 *√*
37 *√*

49 *√*
63

4 *√*

21 *√*
28 *√*

37 *√*
48 *√*

61

5 *√*

31 *√*
39 *√*

49 *√*
61

6 *√*

43 *√*
52 *√*

63

7 *√*

57 *√*
67

8 *√*

73 9

*Figure 7.1: Triangular lattice points*Λ_{1}∪ {0}*along with their distances from the origin (i*≤9).

* Theorem 7.1. Let*r >1. Then, for the triangular lattice pointsΛ

_{1}

*defined by (2.1),*

(7.4) X

x∈Λ1∩Br

1

|x| − 2 r

<0

*holds, where*B_{r} ={x:|x|< r}. Moreover, (7.4) is equivalent to (5.2) and
(7.5)

n−1

X

i=1

1

√i − 2

√n

N^{0}(i)<0
*for*n > 1*with*N^{0}(n)6= 0.

Before presenting the proof of Theorem 7.1, we prove the following lemma.

* Lemma 7.2. Let*k ∈N

*,*r∈R

*with*k < r, andf ≤0

*be convex on*

k− ^{1}_{2}, r
*. Then,*

dre−1

X

i=k

f(i)≤
Z r−^{1}_{2}

k−^{1}_{2}

f(x)dx.

*Proof. Since*k ≤ dre −1< randf is convex on

k−^{1}_{2}, r

, we have (7.6)

dre−1

X

i=k

f(i)≤f(dre −1) +

Z dre−^{3}

2

k−^{1}

2

f(x)dx and

(7.7) (r− dre+ 1)f

r+dre −2 2

≤
Z r−^{1}_{2}

dre−^{3}

2

f(x)dx.

Next, again from the convexity off and0≤ _{r−dre+2}^{dre−r} <1, we have
f(dre −1)≤ 2(r− dre+ 1)

r− dre+ 2 f

r+dre −2 2

+ dre −r

r− dre+ 2f(r).

Thus, considering0< ^{r−dre+2}_{2} ≤1andf ≤0, we have
(7.8) 0≤ −dre −r

2 f(r)≤(r− dre+ 1)f

r+dre −2 2

−f(dre −1).

Then, the required inequality follows by summing up (7.6) and (7.7) side by side and using (7.8). Whenf 6≡0, equality holds iffr∈Nandf is linear.

The proof of Theorem 7.1 comprises 9 steps. As illustrated in Figure 7.2(a), dividing a circular sector at distancerfrom the origin into two regions,A(an equilateral triangle) andB (a circular segment), we shall prove (7.4) onA∪B. By referring to the observations in Remark 3, our approach to the proof is based on simple convexity and monotonicity. The point is to use a mutual elimination between the two terms in (7.4) onB. Figure 7.2(b) illustrates points related toB, which will be explained in step 2 of the proof.

*Proof of Theorem 7.1. Step 1 (equivalence of (7.4),(5.2), and (7.5)). Suppose that (5.2) is sat-*
isfied. Forr >1, choosensuch thata_{n+1} ≥r > a_{n}. Then, considering that#{Λ_{1}∩B_{r}}=n,
we obtain

X

x∈Λ1∩Br

2

r ≥ 2

an+1

#{Λ_{1}∩B_{r}}= 2n
an+1

>

n

X

i=1

1 ai

= X

x∈Λ1∩Br

1

|x|,

*A*

*B*
*s*= ^{√}^{2}_{3}*r*

*r*

(a)

*sv*_{1}
*iv*1

*rv*1

*iv*1+

*i**−*

3(s^{2}*−**i*^{2})

2 + 1

**v**2

*iv*1+

*i**−*

3(s^{2}*−**i*^{2})
2

**v**2

1 **v**1**v**2

(b)

*Figure 7.2: Illustration of (a) regions*A*and*B*and constants*r*and*s, and (b) points related toB.

which gives (7.4). When (7.4) holds, clearly (7.5) holds. Suppose that (7.5) is satisfied. For
eachn ∈ N, letm = an+12 and select the maximumk ≤ n such thatak < an+1. Then, from
the definition ofa_{i} and considering thatPm−1

i=1 N^{0}(i) =k, we haveN^{0}(m)6= 0and

n

X

i=1

2

a_{n+1} − n−k

a_{n+1} = n+k

√m ≥ 2k

√m =

m−1

X

i=1

2N^{0}(i)

√m

>

m−1

X

i=1

N^{0}(i)

√i =

k

X

i=1

1
a_{i} =

n

X

i=1

1

a_{i} − n−k
a_{n+1} ,

which gives (5.2). Consequently, (5.2), (7.4), and (7.5) are all equivalent to each other.

In the following steps, we concentrate on the proof of inequality (7.4).

*Step 2 (division into*A*and*B*). In (2.1), note that each*x ∈ Λ_{1} is given by x = iv_{1} +jv_{2} for
somei∈Nandj ∈ {0, . . . , i−1}, and|x|= [i^{2} −ij +j^{2}]^{1}^{2}. Let

(7.9) s = 2

√3r.

Henceforth, for convenience, we will often usesas well asr. Fori∈N∩[r, s], let

(7.10) k_{i} = i−p

3(s^{2}−i^{2})

2 + 1.

Let

A={(i, j) :i= 1, . . . ,dre −1, j = 0, . . . , i−1},

B ={(i, j) :i=dre, . . . ,dse −1, j =bk_{i}c, . . . , i− bk_{i}c}.

Then, we have

(7.11) {(i, j) :i∈N, j = 0, . . . , i−1,[i^{2}−ij+j^{2}]^{1}^{2} < r}=A∪B.

The proof of (7.11) is given as follows. Ifi∈ {1, . . . ,dre −1}, then[i^{2}−ij +j^{2}]^{1}^{2} < rholds
for allj ∈ {0, . . . , i−1}. Ifi∈ {dre, . . . ,dse −1}, then[i^{2}−ij+j^{2}]^{1}^{2} < ris equivalent to

k_{i}−1 = i−p

3(s^{2}−i^{2})

2 < j < i+p

3(s^{2} −i^{2})

2 =i−k_{i}+ 1