CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION

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1 CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN Abstract. We gve an analytc proof for the Hausdorff convergence of the mdpont or derved polygon teraton. We generalze ths teraton scheme and prove that the generalzaton converges to a regon of postve area and becomes dense n that regon. We speculate on the centrod or derved polyhedron teraton. 1. Background Frst we justfy, as an exercse, a few standard results about nested compact sequences. Then, we examne the mdpont teraton scheme for convex polygons, wth remarks about concave startng condtons and regularty n the lmt. The convergence behavor of the mdpont teraton has been extensvely studed [4]. Our ultmate goal s to defne a generalzaton of the mdpont procedure on the plane and prove smlar convergence results. Ths new teraton wll be characterzed by an ncreasng number of vertces at each step. Our man result s the convergence of these fnte sets of vertces to a dense set of postve area. Fnally, we speculate on the centrod teraton for polyhedra and prove that the lmt s a set of postve volume.. Defntons and Conventons Let d(, ) denote the Eucldean dstance between two ponts n R and the Eucldean norm. Let c( ) denote the convex hull of a set. Denote set closure, wth respect to the standard metrc topology by cl( ) and the open ball of radus ε > 0 about x by B(x, ε). We dentfy a polygon wth a convex hull of a fnte number of affne ndependent ponts on the real plane. Such a convex hull s necessarly bounded and closed. A fnte set of ponts, or vertces, are n general lnear poston f no three dstnct elements of the set are collnear. 3. Compactness of Polygons Our teraton procedures wll deal wth sequences of subsets decreasng under set ncluson. There are many standard results about these nested sequences of compact subsets whch can be appled to polygons on the plane. We prove some here as an exercse for ourselves. Suppose K n } n N s a nested sequence of compact subsets of the plane, such that K n+1 K n for all n N. By the Hene-Borel characterzaton of compact sets n R, each set s equvalently closed and bounded. 1

2 JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN Proposton 3.1. Suppose K n } n N s a nested sequence of nonempty compact sets. Then the ntersecton s nonempty and compact. I = n N K n Proof. Defne a sequence s = s n } n N such that s K j \ K j 1 f and only f = j. Each compact K j \ K j 1 contans fntely many terms s n of the sequence. We can dentfy a convergent subsequence s nk } k N of s wth lmt σ K 1 (and so n all K n ), by compactness. If there exsts an N such that m > N mples σ K m+1 \ K m, then σ would be a lmt pont n K N \ K N 1, contradctng the fact that only fntely many s nk are n K N \ K N 1. So I s nonempty. In addton, I s bounded and closed, snce I K 1 and arbtrary ntersectons of closed sets are closed. Hence, I s compact n R. Besdes the usual area and permeter, the dameter s another useful characterstc of a polygon whch we wll use to prove convergence results. Defnton 3.. Let dam(k) be the dameter of a set K. That s, dam(k) = sup d(x, y). x,y K Proposton 3.3. If K s compact n R, then dam(k) s fnte. Proof. K s bounded, and there exsts a pont p R and M > 0 such that d(x, p) M for all x K. Then, by the trangle nequalty, dam(k) M. Note that the prevous proposton depends only on the boundedness of a compact set K. There s an mmedate connecton between the sequence of dameters of nested compact sets and the dameter of ther ntersecton. Proposton 3.4. Let r 0. If dam(k n ) r for n N, then dam(i) r. Proof. (Sketch) We clam that ths proposton follows from the dscusson n Propostons 3.8 and 3.9 on convergent subsequences of nested compact sets. For brevty, we omt a rgorous proof. Proposton 3.5. Let I = n N K n be the ntersecton of K n. If lm dam(k n ) = 0 then I = x 0 } for some x 0 K 1. Proof. Suppose that I s not a sngleton. So dam(i) 0, and I K n for all n mples lm dam(k n ) > 0. Corollary 3.6. The dameters of the K n converge to the dameter of ther ntersecton; that s, lm dam(k n ) = dam(i).

3 CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION 3 Proof. In Proposton 3.4, let r = lm dam(k n ), snce dam(k n ) s a non-ncreasng sequence. So lm dam(k n ) dam(i). But dam(k n ) bounds dam(i) for arbtrary n snce I K n ; so there s equalty. We can also justfy n nterchangng the lmts n these propostons because dam( ) can be proven to be a contnuous functon wth respect to the Hausdorff metrc, whch we wll now ntroduce. We dentfy a convex polygon wth the convex hull of a fnte set V R of vertces n general lnear poston and seek an approprate sense of convergence. The Hausdorff metrc allows us to defne convergence such that f K n lmts to K, the ponts of K n become arbtrarly close to ther nearest neghbors n K. Defnton 3.7. The Hausdorff dstance of two nonempty compact subsets A and B of a metrc space s defned to be } d H (A, B) = sup sup nf d(a, b), sup nf d(b, a). a A b B b B a A A sequence of nonempty compact subsets K n } converges n the Hausdorff metrc to K f lm d H (K n, K) = 0. The geometrc teratons n the followng sectons produce sequences of nested compact subsets of R. We wll prove convergence results wth respect to ths Hausdorff dstance; therefore, whenever we dscuss the lmt a of sequence of sets, we mean that the sequence comes wthn every ε-ball of the lmt set wth respect to the Hausdorff dstance. (That s, lmts of sequences of sets are taken wth respect to the metrc topology nduced by the Hausdorff dstance on the set of compact subsets of R.) Proposton 3.8. Let K n } n N be a sequence of compact subsets such that K n+1 K n for all n N, and I ther nfnte ntersecton. Then lm d H (K n, I) = 0. Proof. Snce I K n for all n N and so sup y I nf x Kn d(y, x) = 0, we focus on the quantty sup nf d(x, y) = d H(K n, I). x K n y I By compactness, there exst two sequences x n } and y n } such that 1. x n K n \ I. y n I 3. d(x n, y n ) = sup nf d(x, y) x K n y I We can dentfy two convergent subsequences x nk } k N and y nk } k N, wth respectve lmts α and β, such that for all k, d(x nk, y nk ) = sup nf d(x, y). x K nk y I by a subsequence ndex argument. Now, α I necessarly. To see ths, suppose for contradcton that α K n \ I for some n. Then α s a pont of accumulaton and there exsts an ε > 0 such that the open ball B(α, ε) K n \ I contans nfntely

4 4 JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN many ponts of x n }. Then K n \ I contans some x m for m n, contradctng our hypotheses. Therefore d(x nk, y nk ) = nf y I d(x n k, y) d(x nk, α). Passng to the lmt, we have lm k d(x nk, y nk ) = 0. A smlar property holds for countable unons of compact sets: Proposton 3.9. Suppose A n } s a sequence of compact sets such that A n A n+1 and A n K for all n N and some compact K. Let U = n A n. Then lm A n = cl(u). Proof. Snce A n s a subset of K, cl(u) s a bounded and closed set. So dentfy two subsequences a k } and b k } n the compact cl(u) such that 1. a k U. b n A n \ A n 1 3. d(a n, b n ) = sup nf d(x, y) y A n x U Then the same style of proof as n the prevous proposton wth the followng nequalty yelds the result d(b k, a k ) = nf b A k d(b, a k ) d(α, a k ). 4. Mdpont Iteraton Frst solved by the French mathematcan J.G. Darboux, the mdpont polygon problem (sometmes called the derved polygon teraton) has been examned usng dverse technques, ncludng fnte Fourer analyss and matrx products [1] [] [3] [4]. We present an elementary analyss proof of the mdpont polygon problem. For notaton, let P (n) be the set of vertces of the nth convex polygon P (n) n the mdpont teraton. Denote the elements of P (n) by v (n). Use the teraton scheme v (n+1) = 1 (v(n) + v (n) +1 ). Theorem 4.1. (Mdpont teraton): Gven an ntal set of vertces } P (0) = v (0) 1,..., v(0) Q n general lnear poston, the sequence of convex hulls c(p (n) ) produced by mdpont teraton converge n the Hausdorff metrc to the centrod of c(p (0) ). Proof. Wthout loss of generalty, suppose the centrod of P (0) s the orgn. By Proposton 3.8, the Hausdorff lmt lm P (n) s equal to the ntersecton c(p (n) ), whch we know s nonempty and compact. Consder the sequence of dameters D = dam(c(p (n) )) }. Each dameter n N dam(c(p (n) )) = max v (n) v (n) j,j s the arclength of the maxmum lne segment contaned n c(p (n) ).

5 CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION 5 We show by nducton that the maxmum edge length of c(p (n) ) tends to zero as n goes to nfnty. Examne the edge lengths of c(p (1) ), where ndces are taken modulo Q: v (1) v (1) +1 = 1 v(0) v (0) + 1 dam(c(p (0) )) sup v (1) v (1) +1 1 dam(c(p (0) )). Generally, sup v (n) v (n) +1 ( 1 )n 1 dam(c(p (0) )). And the permeter bounds the dameter of a closed convex polygon, so Q dam(c(p (k) )) =1 v (n) v (n) +1 Q(1 )n 1 dam(c(p (0) )). We conclude lm dam(c(p (n) )) = 0. By Proposton 3.5, lm c(p (n) ) = x 0 } for some x 0. But by calculaton, we see that the centrod of P (n+1) s dentcally the centrod of P (n), the orgn. So 0 c(p (n) ) s the lmt of c(p (n) ) }. Now we consder the regularty of the terate polygons. On ths topc, Darboux proves:... ls tendent à devenr semblables à des polygones sem-régulèrs nscrts dans une ellpse. [1]... [the terates] tend to become sem-regular polygons, each of whch s nscrbed n an ellpse. We then reasonably expect the dfference between arbtrary edge lengths of some terate P (n) to tend to zero as n tends to nfnty. Proposton 4.. Let e (n) j of an ntal Q-polygon. Then lm n Proof. We see that: e (n) denote the jth edge length v (n) j e (n) j = 0 for any, j ndces modulo Q. e (n) e (n) j dam(p (n) ) and by the proof of Proposton 3.1, lm dam(c(p (n) )) = 0. v (n) j+1 of the nth terate By the prevous proposton, gven ε > 0, there s a large enough N such that n > N mples for all, j ndces taken modulo Q. e (n) e (n) j < ε 5. Generalzed Mdpont Iteraton In the prevous secton, we consdered a mdpont teraton combnng adjacent vertces of a polygon of the plane. Ths process yelds a natural generalzaton to a mdpont teraton on fnte planar sets whch we call the Generalzed Mdpont Iteraton (GMI). Informally, the GMI on P (0) produces vertex sets P (n) contanng every possble mdpont combnaton of the prevous vertex set P (n 1), and records how many tmes p P (n) occurs as a mdpont by the multplcty functon µ n : P (n) N.

6 6 JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN The GMI s nterestng to study because the number of vertces n each terate grows over tme. Expermentally, ths causes the GMI to converge to a regon of postve area as well as to become dense n ths regon more complcated behavor than the mdpont teraton, whch converges to a sngle pont. Now, we formalze ths new procedure. Defnton 5.1. (Generalzed mdpont teraton): Let } P (0) = p (0) 1,..., p(0) Q R be an ntal set of Q dstnct vertces of a convex polygon. Identfy P (0) wth the vector V (0) (R ) Q wth entres V (0) = (p (0) 1,..., p(0) Q ) Inductvely defne V (n) and P (n) as follows. Let T be any lnear map such that (n 1) p T : V (n 1) j + p (n 1) } : j := V (n) Defne P (n) as the set of entres of V (n). Defne the multplcty µ n (p) of p P (n) as the maxmal number of tmes p occurs n the vector V (n) as an entry. Then the ndexed collecton of tuples: P (k) = (P (n), µ n ) s the generalzed mdpont teraton of the set P (0). Now, for a recursve formula of the multplcty functon µ k, consder any p P (k) for some k. Assocate to p the set M(p) P (k 1) P (k 1) such that } M(p) = (q () 1, q() ) : q() 1 q () and q() 1 + q () = v (J = 1,,..., j v }) where j v s the number of such pars. Next, defne a functon Θ : P (k) N by ( µk 1 ) (p) Θ(p) = f p P (k 1) and µ k 1 (p) > 1 0 else Then, we see that 1 J µ k 1 (q () 1 )µ k 1(q () ) s exactly the number of tmes p occurs as an entry V (k) as the the mdpont of dstnct ponts n V (k 1). Smlarly, Θ(p) s the number of tmes p occurs as an entry as the mdpont of tself n V (k 1). Ths proves the followng proposton: Proposton 5.. Suppose M(p) and Θ(p) are defned as above for p P (k). Then µ k (p) = 1 µ k 1 (q () 1 )µ k 1(q () ) + Θ(p). J The multplcty µ n (p) of a pont p produced by the GMI on the nth teraton determnes the longevty of p. Observe that f µ n (p) > 1, then p wll be an element of V (n+1), and f µ n (p) >, then p wll be an element of V (m) forever, for all m n. J

7 CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION 7 Proposton 5.3. If there exsts n N such that µ n (p) > for some p P (n), then p P (m) for all m n. Proof. µ n (p) > mples p P (n+1). Then, by Proposton 5., ( ) µn (p) µ n+1 (p) µ n (p). Ths mples p P (n+), snce p P (n+) f µ n+1 (p) > 1. Passng to the nductve step, µ k (p) µ k 1 (p) for all k > n; so p P (n) for all k > n. Defnton 5.4. For any n N, f p P (n) satsfes the condtons of Proposton 5.3, then p s a fxed pont of the GMI on P (0). Denote the set of fxed ponts of P (n) on the nth step of GMI by F (n). Recall that n the classcal mdpont teraton, the ntersecton of convex hulls provded the lmt set of the polygons. The fxed ponts descrbed n the prevous proposton wll play an analogous role n defnng the lmtng set of convex hulls of P (n) n the generalzed mdpont teraton. Proposton 5.5. Suppose p P (n) s gven by x+f for some x P (n 1) and f F (n 1). Then p F (n). Proof. Calculatng the multplcty of p, snce µ n 1 (x) 1 and µ n 1 (f) > µ n (p) µ n 1 (x)µ n 1 (f) > In the GMI, fxed ponts always arse quckly f the ntal set P (0) contans at least four dstnct ponts. Proposton 5.6. Suppose P (0) = 4, where denotes cardnalty of a set. Then F (n) 4 for all n >. } Proof. Wthout loss of generalty, ndex P (0) = for J = 1,, 3, 4}. For fxed J, the pont f = 1 4 (3p(0) + p (0) +1 + p(0) + + p(0) +3 ) P (), takng ndces modulo 4, has multplcty at least 3 and s fxed. To see three dstnct constructons of ths pont from the orgnal p (0) by mdponts, consder 1 4 ( 3p (0) = 1 = 1 ( 1 = 1 p (0) ) 1 + p (0) + p (0) 3 + p (0) 4 ( 1 (p(0) 1 + p (0) ) + 1 (p(0) 1 + p (0) 3 )) + 1 (1 (p(0) 1 + p (0) ) + 1 (p(0) 3 + p (0) J 4 ) ) (p(0) 1 + p (0) ) + 1 (p(0) 1 + p (0) 4 )) + 1 (1 (p(0) 1 + p (0) 3 ) + 1 ) (p(0) + p (0) 3 ) ( 1 (p(0) 1 + p (0) ) + 1 (p(0) + p (0) 3 )) + 1 (1 (p(0) 1 + p (0) 3 ) + 1 ) (p(0) 1 + p (0) 4 ) Rangng n J, we have 4 dstnct ponts n F (3), so F (3) 4, and by the persstence of fxed ponts F (n) 4 for all n 3. Corollary 5.7. If P (0) > 4, then F (n) 4 for all n >.

8 8 JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN Proof. If A = p (0) } J P (0) has a cardnalty of 4, then by the prevous proposton, the GMI on P (0) wll produce four fxed ponts f 1,..., f 4 assocated wth A 6. Hausdorff convergence results for the GMI We now prove several convergence results nvolvng the sets of fxed ponts F (n), teraton mdponts P (n), and ther convex hulls c(f (n) ) and c(p (n) ) respectvely. In dong so, we justfy our earler observaton that the GMI converges to a regon of postve area and that the GMI becomes dense n ths regon. We observe that the sequence of convex hulls c(p (n) ) } form a nested sequence of compact n N sets. Proposton 6.1. lm c(f (n) ) exsts, wth respect to the Hausdorff metrc. Proof. By the defnton of fxed ponts, F (n) F (n+1). Ths mples the convex hulls c(f (n) ) c(f (n) ) form an ncreasng sequence of subsets. Lastly, c(f (n) ) c(p (0) ) for all n. So c(f (n) ) } satsfes the condtons of Proposton 3.9. Proposton 6.. lm d H (F (n), c(f (n) )) = 0. Proof. Let x c(f (k) ) for some k. non-collnear ponts Then, by Proposton 5.6, there exst three f 1, f, f 3 } F (k) such that x s n ther convex hull. The mdpont teraton subdvdes F (k) nto four congruent trangles. Inductvely, n teratons of the mdpont procedure subdvdes F (k) nto 4 n congruent trangles. Let T n represent a trangle of the nth subdvson contanng x. By Proposton 3.1, for arbtrary ε > 0, there exsts an N such that for m > N, the vertces v 1, v, v 3 } of T m are wthn ε dstance of the centrod c of T m. Then Ths mples nf d(x, v ) sup d(c, v j ) < ε. j d H (F (n), c(f (n) )) = sup x c(f (k) ) nf d(x, y) < ε. y F (k) Proposton 6.3. lm d H (P (n), F (n) ) = 0. Proof. For notaton, let d(x, A) = nf d(x, y). y A Snce F (n) P (n), the Hausdorff dstance s equal to d H (P (n), F (n) ) = sup x P (n) d(x, F (n) ).

9 CONVERGENCE OF A GENERALIZED MIDPOINT ITERATION 9 Let p P (n+1) arbtrarly. By defnton of the GMI, there exsts at least one par p, q P (n) such that p+q = p. Then, by comparng lengths of smlar trangles created by connectng relevant ponts, d(p, F (n+1) ) 1 d(p, F (n) ) 1 sup x P (n) nf d(x, y). y F (n) Snce p was arbtrary and P (n) and F (n) are fnte sets, we may choose p such that sup d(z, F (n+1) ) = d(p, F (n+1) ). z P (n+1) Ths s precsely the nequalty d H (P (n+1), F (n+1) ) 1 d H(P (n), F (n) ). Proposton 6.4. lm d H (P (k), c(f (k) )) = 0. Proof. By the trangle nequalty, d H (P (k), c(f (k) )) d H (P (k), F (k) ) + d H (F (k), c(f (k) )). The clam then follows from the prevous Propostons 6. and 6.3. Theorem 6.5. The sequence of convex hulls of fxed ponts F (n) and the sequence of convex hulls of P (n) have the same lmt. That s, lm c(f (n) ) = lm c(p (n) ) = c(p (n) ). And consequently lm d H (P (n), c(p (n) )) = 0. Proof. For notaton, let F = lm c(f (n) ) and P = lm c(p (n) ). Seekng contradcton, suppose F P. Then d H (F, P) 0. By Proposton 6., c(f (n) ) } s also a Cauchy sequence. n N By compactness, choose p c(p (N) ) and f c(f (N) ) such that d(p, f) = d H (c(p (N) ), c(f (N) )). Snce F P, d(p, f) s necessarly postve. Now, the mdpont of f and p s n F (N+1) by Proposton 5.5. Denote g = p+f. Then d H (c(f (N) ), c(f (N+1) )) Passng to the lmt, nf d(g, y) 1 d(p, f) > 0. y c(f (N) ) lm N d H (c(f (N) ), c(f (N+1) )) > 0, contradctng the fact that c(f (k) ) } s a Cauchy sequence. Therefore we conclude F = P. Consequently, and d H (P (n), c(p (n) )) d H (P (n), c(f (n) )) + d H (c(f (n) ), c(p (n) )) lm d H (P (n), c(p (n) )) = 0.

10 10 JARED ABLE, DANIEL BRADLEY, ALVIN MOON, AND XINGPING SUN We note that Proposton 6.3 establshes that the GMI converges to ts fxed ponts, certanly a set of postve area. Moreover, Theorem 6.5 establshes that the GMI becomes dense n tself because the ponts themselves lmt to ther convex hull. 7. Centrod teraton for polyhedra The mdpont teraton yelds another generalzaton, the centrod teraton for polyhedra. (We could also call ths the derved polyhedron teraton.) The centrod teraton takes a polyhedron and forms a new polyhedron by takng the centrod of every face to be a new vertex, whch s analogous to takng the mdpont of each sde of a polygon. Expermentally, the centrod teraton shares propertes wth the GMI n that t usually lmts to a regon of postve volume and that t becomes dense n the boundary of ths regon. It may also share the property of regularty n the lmt wth the mdpont teraton, as smulatons suggest that polyhedra lmt to ellpsodal shapes. Unfortunately, smulatng ths process s very computatonally expensve. More work s requred to see f the Hausdorff convergence results for the GMI wll suggest proofs for convergence results for the polyhedron centrod teraton. References [1] Darboux, G., Sur un probleme de geometre elementare, Bulletn des scences mathematques et astronomques e sere, (1878), [] Hntkka, Erc and Xngpng Sun, Convergence of Sequences of Polygons, (014). [3] Ouyang, Charles, A Problem Concernng the Dynamc Geometry of Polygons (013). [4] Schoenberg, I. J., The fnte Fourer seres and elementary geometry, Amer. Math. Monthly, 57 (1950),