Asymptotic Approximation of Convex Curves

Monika Ludwig

Abstract. L. Fejes Tóthgave asymptotic formulae as n → ∞ for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices, where the distance is in the sense of the symmetric difference metric. In this paper these formulae are extended by specifying the second terms of the asymptotic expansions. Tools are from affine differential geometry.

1 Introduction

2 i Let C be a closed convex curve in the Euclidean plane IE and let Pn(C) be the set of all convex polygons with at most n vertices that are inscribed in C. We i measure the distance of C and Pn ∈ Pn(C) by the symmetric diﬀerence metric δS and study the asymptotic behaviour of

S i S i δ (C, Pn) = inf{δ (C,Pn): Pn ∈ Pn(C)}

S c as n → ∞. In case of circumscribed polygons the analogous notion is δ (C, Pn). For a C ∈ C2 with positive curvature function κ(t), the following asymptotic formulae were given by L. Fejes T´oth [2], [3]

Z l !3 S i 1 1/3 1 δ (C, Pn) ∼ κ (t)dt as n → ∞ (1) 12 0 n2 and Z l !3 S c 1 1/3 1 δ (C, Pn) ∼ κ (t)dt as n → ∞, (2) 24 0 n2 where t is the arc length and l the length of C. Complete proofs of these results are due to D. E. McClure and R. A. Vitale [9]. In this article we extend these asymptotic formulae by deriving the second terms in the asymptotic expansions. For a C ∈ C4 these terms are of order 1/n4 and depend on the aﬃne curvature of the curve. Besides the above mentioned results several other asymptotic formulae for the distance of a convex body to its best approximating polytopes are known. In the case of the symmetric diﬀerence metric and inscribed and circumscribed polytopes these formulae were derived by P. M. Gruber in [5], [6], and [8] for any dimension; see also [10]. For detailed information see the surveys [4] and [7].

1 2 Some Tools from Aﬃne Diﬀerential Geometry

S i S c δ (C, Pn) and δ (C, Pn) are invariant with respect to area-preserving affine trans- formations. Therefore, we choose the affine arc length to be the parameter of C. Let C ∈ C2 and let the curvature of C be positive. Then the affine arc length is given by Z t s(t) = κ1/3(τ)dτ, 0 ≤ t ≤ l, (3) 0 where t is the ordinary arc length and κ(t) the curvature of C. The affine length λ of C is Z l λ = κ1/3(τ)dτ. 0 In the following, let x(s) be an affine arclength parametrization of C. The affine curvature of C is then given by

k(s) = |x00(s), x000(s)|, (4) where 0 denotes differentiation with respect to affine arc length. It determines a curve up to an area-preserving affine transformation. Note that

|x0(s), x00(s)| = 1 and |x0(s), x000(s)| = 0. (5)

See [1].

S i 3 Asymptotic Expansion for δ (C, Pn) Using the notions and notation from aﬃne diﬀerential geometry just introduced, we are able to formulate our main result. Theorem 1 Let C be a closed convex curve in IE 2 of class C4 with positive ordinary curvature. Then

3 4 Z λ S i 1 λ 1 λ 1 1 δ (C, Pn) ∼ − k(s)ds + o( ) 12 n2 2 5! 0 n4 n4 as n → ∞. For the proof we need the following two lemmas.

Lemma 1 For 0 ≤ r ≤ s ≤ λ, let F (r, s) be the area of the piece between C and the line segment with end points x(r) and x(s). Then

1 (s − r)3 (s − r)5 ! F (r, s) = − k(r) + o((s − r)5) 2 3! 5! uniformly for all 0 ≤ r ≤ s ≤ λ as (s − r) → 0.

2 Proof. Without loss of generality, let r = 0. Since C ∈ C4, the aﬃne arclength parametrization x(s) of C is of class C3. Let ω be a modulus of continuity function for x000(s); that is, ω is a continuous non-decreasing function on [0, ∞] with ω(0) = 0 and such that

kx000(s) − x000(r)k ≤ ω(|s − r|) for all r, s ∈ [0, λ].

Taylor’s formula yields

00 00 000 x (s) = x (0) + x (0)s + u1(s), (6) where ku1(s)k ≤ sω(s). Integrating (6) twice we see that

s2 x0(s) = x0(0) + x00(0)s + x000(0) + u (s) (7) 2 2 and s2 s3 x(s) − x(0) = x0(0)s + x00(0) + x000(0) + u (s), (8) 2 3! 3 s2 s3 where ku2(s)k ≤ 2 ω(s) and ku3(s)k ≤ 3! ω(s). Since x(s) is the aﬃne arclength parametrization, (4), (5), (6), and (7) imply that s2 1 = |x0(s), x00(s)| = 1 + |x0(0), u (s)| + k(0) + α(s), (9) 1 2 where s2 |α(s)| ≤ µ ω(s) 1 2 and µ1 is a constant which depends only on C. Rewriting (9) in the form

s2 |x0(0), u (s)| = −k(0) − α(s) 1 2 and integrating twice, we ﬁnd that

s3 |x0(0), u (s)| = −k(0) − β(s), (10) 2 3! and s4 |x0(0), u (s)| = −k(0) − γ(s), (11) 3 4! s3 s4 where |β(s)| ≤ µ1 3! ω(s) and |γ(s)| ≤ µ1 4! ω(s). Then combining (7), (8), (10), and (11), we obtain

s2 s4 |x(s) − x(0), x0(s)| = − k(0) + δ(s), (12) 2 4!

3 where 4 |δ(s)| ≤ µ2s ω(s) (13) and µ2 is a constant which depends only on C. Finally, integrating (12) gives

1 Z s 1 s3 s5 Z s ! F (0, s) = |x(s) − x(0), x0(s)|ds = − k(0) + δ(s)ds . 2 0 2 3! 5! 0

Now, noting that [0, λ] is compact and µ2 and ω depend only on C, the last equality together with (13) yields Lemma 1. 2

i Lemma 2 Let Pn ∈ Pn(C), n = 3, 4,..., be a sequence of best approximating polygons of C. Let x(sni), i = 1, . . . , n, be the vertices of Pn. Deﬁne λni = sni − sn,i−1. Then λ 1 λ = + o( ) uniformly in i as n → ∞. ni n n

Proof. Since Pn is a best approximating, the line segment connecting x(sn,i−1) and x(sn,i+1) is parallel to the tangent at x(sni), i. e.

0 |x(sn,i+1) − x(sn,i−1), x (sni)| = 0 for i = 1, . . . , n − 1, (14) see [3]. We proceed by using Taylor’s formula to derive expressions for the λni’s from (14). Let ω be a modulus of continuity function for x000(s). Then for given n and i,

(s − s )2 (s − s )3 x(s) = x(s ) + x0(s )(s − s ) + x00(s ) ni + x000(s ) ni + u (s), ni ni ni ni 2 ni 3! ni where |s − s |3 ku (s)k ≤ ni ω(s − s ). (15) ni 3! ni

Substituting sn,i−1 and sn,i+1 for s and subtracting, we ﬁnd that

x00(s ) x(s ) − x(s ) = x0(s )(λ + λ ) + ni (λ2 − λ2 ) + n,i+1 n,i−1 ni ni n,i+1 2 n,i+1 ni x000(s ) + ni (λ3 + λ3 ) − u (s ) + u (s ). 3! ni n,i+1 ni n,i−1 ni n,i+1 Combining this with (14), we obtain

0 |x(sn,i+1) − x(sn,i−1), x (sni)| =

4 λ2 λ2 = ni − n,i+1 + |x0(s ), u (s )| − |x0(s ), u (s )| = 0. 2 2 ni ni n,i−1 ni ni n,i+1 This can be rewritten as

2 2 0 λni = λn,i+1 + 2|x (sni), uni(sn,i+1) − uni(sn,i−1)| for i = 1, . . . , n − 1. Thus

j−1 2 2 X 0 λni = λnj + 2 |x (snk), unk(sn,k+1) − unk(sn,k−1)| (16) k=i for i < j ≤ n and

i−1 2 2 X 0 λni = λnj − 2 |x (snk), unk(sn,k+1) − unk(sn,k−1)| (17) k=j for 1 ≤ j < i. Summing on j from i + 1 to n in (16) and from 1 to i − 1 in (17), we obtain

n n j−1 2 X 2 X X 0 (n − i)λni = λnj + 2 |x (snk), unk(sn,k+1) − unk(sn,k−1)| j=i+1 j=i+1 k=i i−1 i−1 i−1 2 X 2 X X 0 (i − 1)λni = λnj − 2 |x (snk), unk(sn,k+1) − unk(sn,k−1)|. j=1 j=1 k=j

2 Adding λni to the sum of these equations gives n 2 X 2 nλni = λnj + αni, (18) j=1 where

n j−1 X X 0 αni = 2 |x (snk), unk(sn,k+1) − unk(sn,k−1)| − j=i+1 k=i i−1 i−1 X X 0 −2 |x (snk), unk(sn,k+1) − unk(sn,k−1)|. j=1 k=j

Multiplying (18) by n gives the desired expressions for the λni’s: n 2 2 X 2 n λni = n λnj + nαni, (19) j=1 We will study the behaviour of these equations as n → ∞. For the right-hand side of (19) we will show that

n lim n X λ2 = λ2 (20) n→∞ ni i=1

5 and lim n max |αni| = 0. (21) n→∞ i=1,...,n In order to prove (20) we rewrite the inequalities

n n n λ 1 X 1 X 2 1/2 1 X 3 1/3 = λni ≤ ( λni) ≤ ( λni) , n n i=1 n i=1 n i=1 in the form n n 2 X 2 2 X 3 2/3 λ ≤ n λni ≤ (n λni) . (22) i=1 i=1 As a consequence of Lemma 1,

λ3 F = ni + o(λ3 ) uniformly as λ → 0. ni 12 ni ni Hence, recalling the asymptotic formula (1)

n 3 2 S i 2 X λ lim n δ (C, P ) = lim n Fni = , n→∞ n n→∞ i=1 12 we obtain n lim n2 X λ3 = λ3, (23) n→∞ ni i=1 and (20) follows from (22). 0 Next we establish (21). Deﬁne ν = sup0≤s≤λ kx (s)k. Then (15) gives ν ||x0(s ), u (s )|| ≤ λ3 ω(λ ). (24) ni ni n,i−1 3! ni ni

The deﬁnition of the αni’s implies that

n X 0 |αni| ≤ 2n ||x (snk), unk(sn,k+1) − unk(sn,k−1)||. k=1 Combining this with (24), we have

n n 2 X ν 3 ν 2 X 3 n|αni| ≤ 4n λ ω(λni) ≤ 4 ω( max λni)n λ . (25) ni i=1,...,n ni k=1 3! 3! k=1

S i Since δ (C, Pn) → 0, limn→∞ maxi=1,...,n λni = 0. Hence (21) follows from (23) and (25). By (20) and (21), it follows from (19) that

lim n2 max λ2 = lim n2 min λ2 = λ2, n→∞ i=1,...,n ni n→∞ i=1,...,n ni

6 which proves the lemma. 2 Now we are able to prove the main theorem.

Proof of Theorem 1. First we choose for n = 3, 4,... polygons Qn with vertices λ at the points x(i n ), i = 0, 1, . . . , n, and denote the area of the i-th piece between C and Pn by Fni. Let ε > 0 be chosen. Lemma 1 shows that there is an integer n0 such that 1 1 λ3 1 λ λ5 λ5 ! F ≤ − k(i ) + ε ni 2 3! n3 5! n n5 n5 for all n ≥ n0. Hence 1 λ3 1 λ3 δS(C, Pi ) − ≤ δS(C,P ) − n 12 n2 n 12 n2 n 3 X 1 λ ≤ Fni − 2 i=1 12 n n 3 5 5 ! 3 ! 1 X 1 λ 1 λ λ λ 1 λ ≤ 3 − k(i ) 5 + ε 5 − 2 = 2 i=1 3! n 5! n n n 3! n n 5 5 ! 1 X 1 λ λ λ = (− k(i ) 5 ) + 5 ε = 2 i=1 5! n n n 4 n 5 1 λ 1 X λ λ 1 λ = − 4 k(i ) + 4 ε. 2 5! n i=1 n n 2 n Since ε > 0 was arbitrary, 3 ! 4 Z λ 4 S i 1 λ 1 λ lim sup n δ (C, Pn) − ≤ − k(s)ds. (26) n→∞ 12 n2 2 5! 0

In order to show the opposite inequality, let Pn be a sequence of best approximating polygons. Let x(sni) be the vertices of Pn and let λni = sni − sn,i−1. Choose ε > 0. Then by Lemma 1, there is an integer n0 such that 1 λ3 λ5 ! F ≥ ni − k(s ) ni − ελ5 ni 2 3! n,i−1 5! ni for all n ≥ n0. Using the inequality n n 1 3 1 X 3 X 3 2 λ = 2 ( λni) ≤ λni, n n i=1 i=1 we see that 3 ! n n ! 4 S i 1 λ 4 X 1 X 3 n δ (C, Pn) − 2 ≥ n Fni − λni 12 n i=1 12 i=1 4 n 3 5 3 ! n X λni λni 5 λni ≥ − k(sn,i−1) − ελni − = 2 i=1 3! 5! 3! 4 n 5 ! n X λni 5 = −k(sn,i−1) − ελni . 2 i=1 5!

7 By Lemma 2 λ 1 λ = + o( ) uniformly in i. ni n n Thus

4 n 5 ! 4 n 5 5 ! n X λni 5 n X k(sn,i−1) λ 1 λ −k(sn,i−1) − ελni = − 5 + o( 5 ) − ε 5 2 i=1 5! 2 i=1 5! n n n 4 n 5 1 λ X λ λ = − (k(sn,i−1) ) − ε + o(1) = 2 5! i=1 n 2 4 n 5 1 λ X λ = − k(sn,i−1)λni − ε + o(1). 2 5! i=1 2 Since ε > 0 was arbitrary, we have

3 ! 4 Z λ 4 S i 1 λ 1 λ lim inf n δ (C, Pn) − ≥ − k(s)ds. n→∞ 12 n2 2 5! 0

This together with (26) concludes the proof of Theorem 1. 2

c 4 Asymptotic Expansion for δ(C, Pn) Theorem 2 Let C be a closed convex curve in IE 2 of class C4 with positive ordinary curvature. Then

3 4 Z λ S c 1 λ 1 λ 1 1 δ (C, Pn) ∼ + k(s)ds + o( ) 24 n2 2 5! 0 n4 n4 as n → ∞.

The proof is analogous to that of Theorem 1. As a tool we use the property, that the edges of best approximating circumscribed polygons of C touch C at their midpoints (cf. [3]). Thus instead of (14) we have the equality

I am obliged to Professor Gruber for many valuable suggestions and to Professor Chalk for his helpful comments.

8 References

[1] W. Blaschke, Vorlesungen ¨uber Diﬀerentialgeometrie II, Springer, Berlin 1923

[2] L. Fejes T´oth,Approximation by polygons and polyhedra, Bull. Amer. Math. Soc. 4 (1948), 431-438

[3] L. Fejes T´oth, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Berlin 1953, 1972

[4] P. M. Gruber, Approximation of convex bodies, In: Convexity and its appli- cations (ed. by P. M. Gruber and J. M. Wills), Birkh¨auser, Basel 1983

[5] P. M. Gruber, Volume approximation of convex bodies by inscribed polytopes, Math. Ann. 281 (1988), 229-245

[6] P. M. Gruber, Volume approximation of convex bodies by circumscribed polytopes, In: Applied Geometry and discrete Mathematics. The Victor Klee Festschrift, DIMACS Ser., Vol. 4 (Amer. Math. Soc., Providence, RI) (1991), 309-317

[7] P. M. Gruber, Aspects of approximation of convex bodies, In: Handbook of convex geometry (ed. by P. M. Gruber and J. M. Wills), North-Holland, Amsterdam 1993

[8] P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies II, Forum Math., to appear

[9] D. E. McClure and R. A. Vitale, Polygonal approximation of plane convex bodies, J. Math. Anal. Appl. 51 (1975), 326-358

[10] R. Schneider, Aﬃne-invariant approximation by convex polytopes, Stud. Sci. Math. Hung. 21 (1986), 401-408

Monika Ludwig Abteilung f¨urAnalysis Technische Universit¨atWien Wiedner Hauptstraße 8-10/1142 A-1040 Wien