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Asymptotic Approximation of Convex Curves Asymptotic Approximation of Convex Curves Monika Ludwig Abstract. L. Fejes T´othgave 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.
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