A sufficient condition for the convexity of the area of an isoptic curve of an oval

A Sufficient Condition for the Convexity of the Area of an Isoptic Curve of an Oval.

M. MICHALSKA(*)

ABSTRACT - The problem of convexity of the area function of ana-isoptic of the convex curve C is considered.

1. Introduction.

An a-isoptic C_a of a plane, closed, convex curve C is a set of those points in the plane from which the curve C is seen under a fixed angle p2a. In [1] the authors discussed the problem of convexity of a-isoptic curves. We will study the problem of convexity of the area function.

Letp(t),t[ 0 , 2p] be a support function of the curveC. It is known [1], [2] that the area A(a) of a-isoptic of C is given by formula

A(a)4

0 2p 1

sin²a[p²(t)2p(t1a)(p(t) cosa1p.

(t) sina) ]dt. (1.1)

Letp(t)4a01

!

n41 Q

(ancosnt1bnsinnt) be the Fourier series of the sup- port function. We will assume that p(t)C²[ 0 , 2p] and p(t),p.

(t) L²[ 0 , 2p]. Moreover the curvature radiusR(t)4p(t)1p..

(t)D0 and the Fourier series of R(t)4a₀1

!

n41 Q

(12n²)(a_ncosnt1b_nsinnt) is convergent.

(*) Indirizzo dell’A.: Institute of Mathematics, Maria Curie-Skl^Codowska Uni- versity, pl. M. Curie-Skl^Codowska 1, 20-031 Lublin, Poland; e-mail: daisyHgo- lem.umcs.lublin.pl

Let f(x)A_n

!

4 2Q Q

c_ne^inx, g(x)A_n

!

4 2Q Q

d_ne^inxL² where c_2n4c_n are complex numbers then we have ([5], p. 37, theorem 1.12)

THEOREM 1.1. Suppose that f and g are in L²and have coefficients c_n and d_n respectively. Then

1 2p

0 2p

f(x2t)g(t)dt4_n

!

4 2Q Q

c_nd_ne^inx (1.2)

for all x, and the series on the right converges absolutely and uniformly.

In particular 1 2p

0 2p

f(x1t) f(t)dt4_n

!

4 2Q Q

Nc_nN²e^inx, (1.3)

1 2p

0 2p

f(t)g(t)dt4_n

!

4 2Q Q

c_nd_2n, (1.4)

1 2p

0 2p

Nf(t)N²dt4_{n4 2Q}

!

If f(t)4a01

!

n41 Q

(a_ncosnt1b_nsinnt) is a real valued function and g4f8 then we have for all xR

0 2p

f²(t)dt42pa₀²1p

!

n41 Q

(a_n²1b_n²) , (1.6)

0 2p

f(t1x) f(t)dt42pa₀²1p

!

n41 Q

(a_n²1b_n²) cosnx, (1.7)

0 2p

f(t1x) f8(t)dt4p

!

n41 Q

n(a_n²1b_n²) sinnx (1.8)

which, with formula (1.1), give the following formula for the area of the a-isoptic

(1.9) A(a)4 4 1

sin²a

k

^!

g

^!

h

2sinaQp

!

n41 Q

n(a_n²1b_n²) sinna

l

4 p

sin²a

k

^!

l

4 p

sin²a

k

^!

(a_n11² 1b_n11² )

g

2 cosna

hl

2. The main result.

THEOREM 2.1. Let p(t)4a01

!

n41 Q

(a_ncosnt1bnsinnt)be the Fouri- er series of the support function of C.If Fourier coefficients of p(t)are such that the inequality holds

a₀²F

!

n40 Q

(n(n12 ) )²(a_n11² 1b_n11² ) (2.1)

then A(a) is a convex function.

PROOF. We will show thatA9(t)F0 . From the above considerations we get

A8(a)

p 42a₀² ( 12cosa)² sin³a 1 1

!

n40 Q

(a_n²₁₁1bn211) 1

sin³a

y

22 cosa

g

2 cosna

hl

and by some calculations (2.2) A9(a)4 p

sin⁴a

m

1

!

n40 Q

(a_n12 1

1b_n12 1)[n³sin³asin (n11 )a13n²sin²acosna2 26nsinacosasinna12( 112 cos²a)( 12cosna) ]

n

If the condition (2.1) holds, then A9(a)

p F 1

sin⁴a

!

n40 Q

(a_n²₁₁1b_n²₁₁)]2n²(n12 )²( 12cosa)²( 22cosa)1 1n³sin³asin (n11 )a13n²sin²acosna26nsinacosasinna1 12( 112 cos²a)( 12cosna)(

4 1 sin⁴a

!

n40 Q

(an211

1bn211)Mn(a), where

(2.3) Mn(a)42n²(n12 )²( 12cosa)²( 22cosa)1n³sin³asin (n11 )a1 13n²sin²acosna26nsinacosasinna12( 112 cos²a)( 12cosna) for abbreviation. One can see that M_n( 0 )40 for nN andM₀(a)40 . We will show that M_n8(a)F0 . We have

M_n8(a)42(n⁴14n³14n²) sina( 12cosa)( 523 cosa)1

1n⁴sin³acos (n11 )a14n³sin³acos (n11 )a1 18nsin²asinna28 sinacosa( 12cosna) . Ifa[ 0 , p) then sinaF0 andNsinnaN Gnsinaandn⁴14n³F5n²for any positive integer n. Moreover, we have

M_n8(a)Fn⁴sina( 12cosa)[ 2( 523 cosa)2( 11cosa) ]1

14n³sina( 12cosa)[ 2( 523 cosa)2( 11cosa) ]1 18n²sina( 12cosa)( 523 cosa)2

28n²sin³a28 sinacosaQ2 sin²na 2 F F(n⁴14n³) sina( 12cosa)[ 927 cosa]1

18n²sina( 12cosa)[ 523 cosa212cosa]2 216n²sinacosasin² a

2 F5n²Q2 sinasin² a

2( 927 cosa)216n²sinacosasin² a 2 4 42n²sinasin² a

2[ 45235 cosa28 cosa]4

42n²sinasin² a

2( 45243 cosa)F0 .

The functionM_n(a) is not decreasing for eacha[ 0 ,p) and M_n( 0 )40 . This means thatM_n(a)F0 for eacha[ 0 ,p) and each positive integer n and M0(a)40 . This completes the proof of the inequality

A9(a)

p F 1

sin⁴a

!

n40 Q

(a_n²₁₁1b_n²₁₁)M_n(a)F0 for each a[ 0 ,p). r

EXAMPLE 1. Let p(t)4a1bcos 3 t , a,bD0 then the curvature radiusR(t)4a28bcos 3t. If aD8b then the curve C is convex. From (2.1) the area function A(a) is convex if a²F( 8b)².

EXAMPLE 2. Let p(t)4a1bcos 3t1ccos 5t then from (2.1) a²F F( 8b)²1( 24c)²andR(t)4a28bcos 3t224ccos 5tD0 foraD8NbN 1 124NcN. We have

a²D( 8NbN 124NcN)²D( 8b)²1( 24c)².

For any convex curve C with support function p(t)4a1bcos 3t1 1ccos 5t the area function A(a) is convex.

EXAMPLE 3. Consider a support function p(t)4 ¹

12asint, t[ 0 , 2p). Then the curvature radius of the corresponding oval is R(t)4^123asint12a

2

( 12asint)³ F0 for 0E NaN E¹₂. We will find the Fourier series of p(t)4a₀1

!

n41 Q

(a_ncosnt1b_nsinnt).

1

12asint4

!

n40 Q

(asint)ⁿ, where

sin^mt4

. ` / `

´

(21 )^k 2^2k

y

!

j40 k21

g

h

g

h ^z

(21 )^k 2^2k

y !

j40

k

g

h

^z

hence we have 1

12asint 4

!

k40 Q

a^2k

g

h

2^2k 1

!

k40 Q

(21 )^k11cos 2(k11 )t

!

j4k Q a^2j12

2^2j11

g

h

1

!

k40 Q

(21 )^ksin ( 2k11 )t

!

j4k Q a^2j11

2^2j

g

h

The Fourier coefficients of p(t) are equal to

a_n4

. ` / `

´

k

!

a^2k

g

h

2^2k for n40 (21 )^k11

!

j4k Q a^2j12

2^2j11

g

h

0 for n42k11 ,

bn4

. /

´

0 for n42k (21 )^k

!

j4k Q a^2j11

2^2j

g

h

Using the inequalities

k2pn

g

h

g

h

g

h

g

h

we have

g

h

2^2j11 E 2

kpkj11 G 2 kpkk11 ,

g

h

2^2j E 2

kpkj G 2

kpkk , for j4k,k11 ,R

and

Na_2k12N 4

!

j4k Q a^2j12

2^2j11

g

h

^!

2^2j11

g

h

^!

Q 2a^2j12

k

4 2a^2k12

k

Nb_2k11N 4

!

j4k Q a^2j11

2^2j

g

h

^!

2^2j

g

h

G

!

j4k Q 2a^2j11

kpk 4 2a^2k11 kpk

1 12a² . We can estimate the right hand side in the formula (2.1)

n

!

(n(n12 ) )²(an211

1bn211

)4

4

!

k40 Q

( 2k11 )²( 2k13 )²a_2k²₁₂1

!

k40 Q

( 2k)²( 2k12 )²b_2k²₁₁E

E

!

k40 Q

( 2k11 )²( 2k13 )²

u _k

11 ) 1 12a²

v

1

!

k40 Q

( 2k)²( 2k12 )²

g

h

G 1

p( 12a²)²

!

k40 Q

[

]

4 1

p( 12a²)²

!

k40 Q

[ (a^4k15)R24(a^4k14)9 23(a^4k13)8 23a^4k121

1(a^4k17)R24(a^4k16)9 23(a^4k15)8 23a^4k14]4

8a⁴( 924a²1a⁴) p( 12a²)⁶ .

On the other hand

a₀²4

u !

k40 Q

a^2k

g

h

2^2k

v

u

!

k41 Q

a^2k

g

h

2^2k

v

Comparing both sides we obtain that if 0E NaN E

o

PROPOSITION 2.1. Let p(t)be the support function of the curve C.If there existseD0such that fora[ 0 ,e]Fourier coefficients of p(t)sat- isfy inequalities

(2.4)

!

n40 Q

(a_n11² 1b_n11² )[n³sin³asin (n11 )a13n²sin²acosna2 26nsinacosasinna12( 112 cos²a)( 12cosna) ]F0 and

a₀²FC(e)

!

n40 Q

(n³13n²16n)(a_n²₁₁1b_n²₁₁) , (2.5)

where C(e)4_{2( 1} ¹

2cose)²( 22cose) then A(a) is a convex function.

PROOF. Ifa[0,e] then from the assumption (2.4) we haveA9(a)F0.

For a(e,p) we have

n³sin³asin (n11 )a13n²sin²acosna2

26nsinacosasinna12( 112 cos²a)( 12cosna)F 2n³23n²26n and if a₀ satisfy (2.5) then we obtain

A9(a)

p 42a₀²( 12cosa)²( 22cosa)

sin⁴a 1

1 1 sin⁴a

!

n40 Q

(an211

1bn211)[n³sin³asin (n11 )a1

13n²sin²acosna26nsinacosasinna12( 112 cos²a)( 12cosna) ] F 1

sin⁴a

y

!

n40 Q

(n³13n²16n)(an211

1bn211

) 1 C(e)2 2

!

n40 Q

(a_n12 1

1b_n12 1)(n³13n²16n)

l

In particular, if the Fourier coefficients of p(t) are such that )n0N (nDn0 a_n4b_n40

then p(t) satisfy (2.4) for some e0D0 . In fact if we put Nn11(a)4 1

sin⁴a[n³sin³asin (n11 )a1

13n²sin²acosna26nsinacosasinna12( 112 cos²a)( 12cosna) ] in (2.2) we have lim

aK0¹

Nn11(a)4ⁿ

2(n12 )²

4 D0 for each 0EnGn0 and N₁f0 .N_n11(a) is a continuous function and therefore existsenD0 such that N_n11(a)D0 fora[ 0 ,en] we definee04min]e1,e2,R,en₀(. If for e0 conditon (2.5) holds then A(a) is a convex function.

The author would like to thank the Referee of this paper for his re- marks which improve this paper.

R E F E R E N C E S

[1] W. CIES´LAK- A. MIERNOWSKI - W. MOZGAWA, Isoptics of a Closed Strictly Convex Curve, Lect. Notes in Math., 1481(1991), pp. 28-35.

[2] W. CIES´LAK- A. MIERNOWSKI - W. MOZGAWA, Isoptics of a Closed Strictly Convex Curve II, Rend. Sem. Mat. Univ. Padova,96 (1996), pp. 37-49.

[3] H. GROEMER, Geometric applications of Fourier series and spherical har- monics, Encyclopedia of Mathematics and its Applications, Cambridge Univ.

Press, Cambridge, 1996.

[4] M. A. HURWITZ,Sur quelques applications géométriques des séries de Fouri- er, Ann. Sci. Ecole Norm. Sup., 19 (1902), pp. 357-408.

[5] A. ZYGMUND, Trigonometric series, Cambridge Univ. Press, vol. I, II, 1968.

Manoscritto pervenuto in redazione il 9 gennaio 2003.