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On one extremal property of a regular simplex 1 1 Vladislav Babenko, Yuliya Babenko, 0 Nataliya Parfinovych, and Dmytro Skorokhodov 2 n a J Abstract 3 Inthispaper,weshowthattheLp-errorofasymmetriclinearapproximation 1 of the quadratic function Q(x)= d x2 on simplices in Rd of fixed volume is j=1 j minimized on regular simplices. ] P A N 1 Introduction . h t Oneimportantproblemingeometryistostudytheapproximation(inaspecified a metric) of smooth convex bodies by various polytopes. For instance, questions m of approximating convex bodies by inscribed or circumscribed polytopes, by [ polytopes with fixed number of vertices or faces, by polytopes of best approxi- 1 mation, etc. have been studied in this direction. After some occasional results v in the plane, the book of Fejes Toth [27] was the first to provide a large num- 6 ber of problems, ideas and results on polytopal approximation in dimensions 9 twoandthree,concentratingspecificallyonextremalpropertiesofregularpoly- 4 2 topes. Manyextensionshavebeenmadeafterwardstohigherdimensions,other . metrics, etc. (see [7,8,16,17] and references therein). 1 On the other hand, the question on approximation of functions, defined on 0 1 a polytope, by piecewise linear functions, generated with the help of triangula- 1 tions, in L -metrics is of great importance in approximationtheory. The prob- p v: lems of interpolation, best and best one-sided approximation of functions by i linear splines (i.e., piecewise linear functions) have been considered. The ques- X tion of finding optimal adaptive triangulations, i.e., triangulations that depend r on the function being approximated (see, for instance, [13,14]), is of particular a interestforapplications. Inordertoconstructadaptivetriangulationsmanyau- thors took into account the Hessian of the function (or curvature of its graph) (see [5,6,10,12,18,24]). Note that the construction of the best (in a specified sense) polytope for an individual convex body, or construction of the best triangulation for approximation of a specific individual function, is an extremely difficult problem. The above-mentionedpapershavedealtonlywith asymptoticallyoptimal sequences of polytopes or asymptotically optimal sequences of triangulations. One possible method to construct an asymptotically optimal sequence of polytopes or triangulations is as follows. As the first step, we construct an in- termediate approximation of the convex body surface (or the function, respec- 1 tively) by piecewise quadratic surface (function), and then solve the problem of approximatingthe piecewise quadratic surface (function) by piecewise linear ones. The latter, in turn (at least for d = 2), is equivalent to solving the following optimization problem (we shall give its statement for approximation of functions). Let a quadratic function Q and a simplex T in Rd of unit volume be given. We shall consider the best L -approximation of function Q by linear functions p defined on T (or the best one-sided approximation, which coincide with the deviation of interpolant for positively definite Q), and the problem is to find a simplexT∗,forwhichthecorrespondingerrorisminimal. (Theknownsolutions of this problem are listed in Section 3.) Therefore, in a number of questions of geometry and approximationtheory, itisimportanttofindasimplexoffixed(unit)volumesuchthattheerrorofthe best approximation of a given quadratic function on this simplex in a specified metric or the best approximation with constraints (for instance, one-sided) is minimized. In approximation theory, there exists a tool to view both the problem of finding the best approximation without constraints and the problem of finding the best approximation with constraints “under one umbrella”. The latter can be viewed as the best approximation in the spaces with asymmetric norm or so-called (α,β)-approximation (see, for example, [1,2,20]), when positive and negative parts of the difference between function and the approximant are “weighted” differently. Such type of approximations are of a separate interest, since they can be considered as the problems of approximation with non-strict constraints (see below for more precise statements), when constraints are allowed to be violated, but the penalty for the violation is introduced into the errormeasure. Wethinkthatsuchamethodcouldalsobeinterestingforcertain geometric problems. Therefore, the purpose of this paper is to prove the optimality of a regular simplex in the problem of minimizing (over the simplices unit volume) the best (α,β)-approximation in L -metric of quadratic function p d x2 by linear functions. Note that with the help of linear transformations j=1 j the solution of this problem allows us to obtain the solution of analogous opti- P mization problems for an arbitrary positive definite quadratic form. The paper is organized as follows. Section 2 contains definitions, notations and rigorous statements of the problem we study and closely related problems. PreviousresultsandthemainresultofthepaperaregiveninSection3. Section4 is devoted to the proof of the main result. 2 2 Notations, definitions and statements of the problems Letd∈NandletRdbethespaceofpointsx=(x ,...,x ). Everypointx∈Rd 1 d determines(andisdeterminedby)therow-vectorwithcoordinates(x ,...,x ), 1 d and we shall reserve the notation x for such a vector. The Euclidean distance between points a,b∈Rd is defined as usually by 1/2 d ka−bk := (a −b )2 . 2 j j   j=1 X   For a row-vectorx, let xt be the column-vector transponent to x. For a square matrix J, we denote its transponent matrix by Jt. For a measurable bounded set G⊂Rd, let L (G), 1≤p≤∞, be the space p ofmeasurableandintegrableinthe powerp(essentiallyboundedifp=∞)and functions f :G→R endowed with the usual norm 1 |f(x)|pdx p if 1≤p<∞, kfkLp(G) :=(essGsup{|f(x)|:x∈G} if p=∞. (cid:0)R (cid:1) Let f ∈ L (G) and let a locally compact subset H ⊂ L (G) be given. Set p p E(f;H) to be the best approximation of the function f by H in the L - Lp(G) p norm, i.e., E(f;H) :=inf{kf −uk : u∈H}. Lp(G) Lp(G) In addition, set E±(f;H) :=inf{kf −uk : ±u≤±f, u∈H}. (2.1) Lp(G) Lp(G) Quantity (2.1) is called the best approximation from above (E−(f;H) ) Lp(G) or below (E+(f;H) ) of the function f by the subset H in the L -norm. Lp(G) p The quantities E−(f;H) and E+(f;H) are called the best one-sided Lp(G) Lp(G) approximations. For α,β >0 and f ∈L (G), let p |f(x)| :=αf (x)+βf (x), α,β + − where g (x):=max{±g(x);0}. Define the asymmetric L -norm as follows: ± p kfk :=kαf +βf k Lp;α,β(G) + − Lp(G) 1 |f(x)|p dx p if 1≤p<∞, = G α,β (cid:16)esRssup{|f(x)|α,β(cid:17):x∈G} if p=∞. Asymmetricnormsinconnectionwithvariousproblemsinapproximationtheory were considered in papers [1, 4, 21] and books [19, 20, 22]. By 3 E(f;H) denote the best (α,β)-approximation [1,20] of the function f by p;α,β H in the L -norm, i.e., p E(f;H) =inf{kf −uk : u∈H}. Lp;α,β(G) Lp;α,β(G) Note that for α=β=1, we have E(f;H) =E(f;H) . Babenko [1] Lp;1,1(G) Lp(G) proved that the following limit relations hold (see also [20, Theorem 1.4.10]): lim E(f;H) =E+(f;H) , β→+∞ Lp;1,β(G) Lp(G) (2.2) lim E(f;H) =E−(f;H) . α→+∞ Lp;α,1(G) Lp(G) This allows us to include the problem of the best unconstrained approximation andtheproblemofthebestone-sidedapproximationintothefamilyofproblems of the same type, and consider them from a general point of view (for more on this motivation, see [2,3]). Because of the relation kf −ukp =kf −ukp+(βp−1)k(f −u) kp, β >1, p;1,β p − p the problem of the best (1,β)-approximationcan be considered as the problem of the best approximation with non-strict constraint f ≤ u. This constraint is allowed to be violated, but the penalty (βp−1)k(f −u) kp − p for its violation is introduced into the error measure. In what follows, we shall allow the value +∞ for α or β, in that case identifying E(f;H) with Lp;α,β(G) the corresponding one-sided approximation. Let S (G):={g(x)=axt+c : a∈Rd, c∈R, x∈G}. 1 ThespaceS (G)willbethemainapproximationsetinthispaper. LetalsoT = 1 {t1,...,td+1} be the d-dimensional simplex with vertices tj, j = 1,...,d+1. We shall consider the following optimization problem. Let Q(x)=xxt, and for T ⊂Rd, set E(Q;S (T)) σ (T):= 1 Lp;α,β(T), p;α,β;d |T|1+p1 where |T| stands for the d-dimensional volume of the simplex T. The purpose of this paper is to solve Problem 1. Find σ :=infσ (T) (2.3) p;α,β;d p;α,β;d T and describe all simplices T, for which the infimum in the right-hand part of (2.3) is achieved. A solution to Problem 1 will allow to solve the following related problems. 4 Problem 2. Find E(Q;S (T)) σ :=inf 1 Lp(T) (2.4) p;d T |T|1+p1 and describe all simplices T, for which the infimum in the right-hand part of (2.4) is achieved. Problem 3. Find E±(Q;S (T)) σ± :=inf 1 Lp(T) (2.5) p;d T |T|1+p1 and describe all simplices T, for which the infimum in the right-hand part of (2.5) is achieved. 3 History and the main result Note that the quantity E+(Q;S (T)) coincides with the error of linear 1 Lp(T) interpolation of the quadratic function Q(x) at the vertices of the simplex T, and the quantity E−(Q;S (T)) coincides with the error of tangential in- 1 Lp(T) terpolation of Q(x) on the simplex T. In view of formulas (2.2), we have σ+ = lim σ and σ− = lim σ . p;d β→+∞ p;1,β;d p;d α→+∞ p;α,1;d The quantity σ+ has been considered for d = 2 in connection with the p;d problem of finding the best triangulation △ consisting of N triangles of the N set G ⊂ R2, provided that the L -error of interpolation at the vertices of △ p N of convex function f is minimized. The first attempt to find σ+ is due to D’Azevedo and Simpson [12], who p;d computedσ+ . Tothebestofourknowledge,theprogressonthisproblemcan ∞;2 be outlined as follows: (1) d=2, p=∞ [12]; (2) d≥2, p=∞ [26]; (3) d=3, p=2 [9]; (4) d=2, p=1 [7]; (5) d=2, p=2 [25]; (6) d≥2, p∈N [10]; (7) d=2, p∈(0,∞) [6]; (8) d≥2, p∈(1,∞) [11]. Remark 3.1. Note that σ+ for d=2 and p∈(1,∞) was independently found p;d by Chen [11] and Babenko, et al. [6]. 5 Remark 3.2. By L -error in the case p ∈ (0,1), we understand the following p expression: 1 p E(f;H) :=inf |f(x)−u(x)|pdx : u∈H . Lp(G) ((cid:18)ZG (cid:19) ) Remark 3.3. Infimum in Problem 3 is achieved only on regular simplices. To the best of our knowledge, Problem 2 was solved only in the case d=2, p = 2 and α = β = 1 by Nadler [23,24]. In the next section, we shall give the solution of Problem 1 for all α,β >0, 1≤p≤∞ and d∈N. Let T be a regular simplex of unit volume in Rd. The main result of our 0 paper is the following: Theorem 3.1. Let α,β >0, d∈N and 1≤p≤∞. Then σ =σ (T ). p;α,β;d p;α,β;d 0 In view of (2.2) we obtain the following statements. Corollary 3.1. Let d∈N and 1≤p≤∞. Then E(Q;S (T )) σ = 1 0 Lp(T0). p;d |T0|1+p1 Corollary 3.2. Let d∈N and 1≤p≤∞. Then E±(Q;S (T )) σ± = 1 0 Lp(T0). p;d |T0|1+p1 Recall that E+(Q;S (T )) is the error of linear interpolation of 1 0 function Q. The next section is devoted to the proof of this theorem. 4 Proof of the main result Let α,β > 0 be fixed throughout this section. Note that the value of the quantity E(Q;S (T)) is independent of translations of the simplex T 1 Lp;α,β(T) and its volume. For simplices T,T′ ⊂Rd, we shall write T =T′ if there exists a motion F of the space Rd such that F(T) = T′, and we shall write T 6= T′ otherwise. The proofofthe maintheoremconsistsoftwopartscontainedinthe following two lemmas. Lemma4.1. LetT beanarbitraryd-dimensionalsimplexofunitvolume. Then there exists a constant C >0, independent of T, such that σ (T)≥C(diamT)2. p;α,β;d 6 Lemma 4.2. If T, T 6=T , is a simplex of unit volume in Rd then there exists 0 a simplex T∗ ⊂Rd of unit volume such that σ (T)>σ (T∗). p;α,β;d p;α,β;d Indeed,inviewofLemma4.1,thereexistsanoptimald-dimensionalsimplex T′ of unit volume such that σ = σ (T′). Then, Lemma 4.2 gives p;α,β;d p;α,β;d T′ =T . 0 Proof of Lemma 4.1. Let T := T = {t1,t2,...,td+1} be a simplex of unit d volume. Assume that kt1−t2k = diamT . In addition, for j = 1,...,d−1, 2 d let T ={t1,t2,...,tj+1} be a simplex in Rj. j First, let us consider the case 1 ≤ p < ∞. For j = 2,...,d, by h denoting j the length of the height from the vertex tj of the simplex T to the simplex j T . For every a=(a ,...,a )∈Rd, let a′ =(a ,...,a ). Then, j−1 1 d 1 d−1 σp (T )= inf |xxt−axt−c|p dx p;α,β;d d a∈Rd,c∈RZTd α,β hd = inf |u2+yyt−a u−a′yt−c|p dydu a∈Rd,c∈RZ0 ZhudTd−1 d α,β hd u 2p+(d−1) ≥ σp (T )du h p;α,β;d−1 d−1 Z0 (cid:18) d(cid:19) h = d σp (T ). 2p+d p;α,β;d−1 d−1 Proceeding by induction on d, we verify that h h ...h σp (T )≥ 2 3 d σp (T ) p;α,β;d d (2p+2)(2p+3)···(2p+d) p;α,β;1 1 h h ...h ≥Cp· 2 3 d ·(diamT )2p+1 =Cp(diamT )2p, d! d d where Υ is some positive constant independent of the simplex T . d Let us turn to the case p=∞. In this case, we obtain σ (T )≥σ (T ) ∞;α,β;d d ∞;α,β;1 1 = inf sup |u2−ku−c| ≥Υ(diamT )2. α,β d k∈R,c∈Ru∈[0,diamTd] 7 Proof of Lemma 4.2. Let T ={w1,w2,t1,...,td−1}6=T be a simplex of unit 0 volume. Without loss of generality, we may assume that kw1−t1k 6= kw2− 2 t1k . Clearly, we can always choose the coordinate system in Rd so that the 2 vertices of T have the following coordinates: w1 =(−δ,0,0,...,0), w2 =(δ,0,0,...,0), where δ := 1kw1 −w2k , and the remaining vectors t1,t2,...,td−1 have the 2 2 following coordinates: t1 =: ( b a 0 0 ... 0 0 ), 1 1,1 t2 =: ( b a a 0 ... 0 0 ), 2 1,2 2,2 t3 =: ( b a a a ... 0 0 ), 3 1,3 2,3 3,3 . . . . . . . . . . . . . . . . . . . . . td−2 =: ( b a a a ... a 0 ), d−2 1,d−2 2,d−2 3,d−2 d−2,d−2 td−1 =: ( b a a a ... a a ). d−1 1,d−1 2,d−1 3,d−1 d−2,d−1 d−1,d−1 Note that in view of our assumption, b 6=0. In addition, it can be easily seen 1 that a 6=0 for all j =1,...,d−1. j,j Let b:=(b ,...,b ) 1 d−1 and a a a ... a a 1,1 1,2 1,3 1,d−2 1,d−1 0 a a ... a a 2,2 2,3 2,d−2 2,d−1   0 0 a ... a a 3,3 3,d−2 3,d−1 A:= .. .. .. .. .. .. .  . . . . . .     0 0 0 ... ad−2,d−2 ad−2,d−1  0 0 0 ... 0 ad−1,d−1 Since the matrix Ais non-singular, set  y=(y ,...,y ):=bA−1. (4.1) 1 d−1 Let I be the identity matrix of size (d−1)×(d−1). In addition, set R:= yty+I. ItcanbeeasilyseenthatthematrixRispositivedefinite. Therefore,in viewoftheCholeskydecomposition(astandardtechniqueinnumericalanalysis, whose description can be found, for instance, in [15]), there exists an upper triangular matrix U=(u ) such that k,j 1≤k,j≤d−1 R=UtU. Moreover,for every j =1,...,d−1, we have u = Dj , where j,j Dj−1 q 1+y2 y y y y ... y y y y 1 1 2 1 3 1 k−1 1 k y y 1+y2 y y ... y y y y  1 2 2 2 3 2 k−1 2 k  y y y y 1+y2 ... y y y y 1 3 2 3 3 3 k−1 3 k D0 :=1, Dk :=det ... ... ... ... ... ... ,   y1yk−1 y2yk−1 y3yk−1 ... 1+yk2−1 yk−1yk  y1yk y2yk y3yk ... yk−1yk 1+yk2  (4.2) 8 for all k =1,...,d−1. Consequently, u >0 for every j =1,...,d−1. j,j Let Q = (q ) be the diagonal matrix such that q = u , j = k,j 1≤k,j≤d−1 j,j j,j 1,...,d−1. We define U:=Q−1U. Then U is the unit upper triangular matrix. Therefore, R=UtQ2U. Set M:=UA. Denote the elements of M by m , i.e., M = (m ) . Note that for k,j k,j 1≤k,j≤d−1 every j =1,...,d−1, it follows that m =a . j,j j,j Let us now consider the simplex T ={w1,w2,t1,...,td−1}, whose vertices have the following coordinates: e e e t1 := ( 0 a 0 0 ... 0 0 ), 1,1 t2 := ( 0 m a 0 ... 0 0 ), 1,2 2,2 et3 := ( 0 m m a ... 0 0 ), 1,3 2,3 3,3 e.. .. .. .. .. .. .. . . . . . . . e td−2 := ( 0 m m m ... a 0 ), 1,d−2 2,d−2 3,d−2 d−2,d−2 td−1 := ( 0 m m m ... m a ). 1,d−1 2,d−1 3,d−1 d−2,d−1 d−1,d−1 e Obviously, the volumes of simplices T and T coincide. e Now we shall construct the linear transformation S : Rd → Rd such that S(T)=T. To that end, set e h:=bM−1 ∈Rd−1, e and define the linear transformation S with the help of the matrix 1 h 0 S=. . . (U)−1 .   0      Note that detS = 1. Then for every a = (a ,a ,...,a ) ∈ Rd and c ∈ R, it 1 2 d follows that L:=kxxt−axt−ckLp;α,β(T) =kvStSvt−aSvt−ckLp;α,β(Te). (4.3) Let us consider the simplex T ={w1,w2,t1,...,td−1} such that t1 := ( −b a 0 0 ... 0 0 ), 1 1,1 b b b t2 := ( −b a a 0 ... 0 0 ), 2 1,2 2,2 bt3 := ( −b a a a ... 0 0 ), 3 1,3 2,3 3,3 b.. .. .. .. .. .. .. . . . . . . . b td−2 := ( −b a a a ... a 0 ), d−2 1,d−2 2,d−2 3,d−2 d−2,d−2 td−1 := ( −b a a a ... a a ). d−1 1,d−1 2,d−1 3,d−1 d−2,d−1 d−1,d−1 b b 9 ItcanbeeasilyverifiedthatthelineartransformationS :Rd →Rd definedwith the help of the matrix 1 −h b 0 S=.. (U)−1 .   b 0      transforms the simplex T into the simplex T. Therefore, for a := (−a , 1 a ,...,a ), we obtain 2 d e b b L:=kxxt−axt−ck b =kvStSvt−aSvt−ck e . (4.4) Lp;α,β(T) Lp;α,β(T) Dube to the symbmetry of simplices Tb band Tb,bwe obtain L = L. Then from (4.3) and (4.4), we derive that b b 1 1 L= (L+L)≥ v(StS+StS)vt−(aS+aS)vt−2c . (4.5) 2 2 e (cid:13)(cid:13) h i(cid:13)(cid:13)Lp;α,β(T) Note that b (cid:13)(cid:13) b b bb (cid:13)(cid:13) 1 h 1 0 ... 0 0 2D:=StS+StS = ht (U)−1 t  ... (U)−1 (cid:18) (cid:19)  b b (cid:2) (cid:3)  0   1 −h 1 0 ... 0 0 + −ht (U)−1 t ... (U)−1 (cid:18) (cid:19)  0  (cid:2) (cid:3)   1 0 ... 0   0 = 2.. hth+[(U)−1]t(U)−1. .   0      Since h=bM−1 =bA−1(U)−1, we have hth+[(U)−1]t(U)−1 =[(U)−1]t[(A−1)tbtbA−1+I](U)−1 =[(U)−1]tR(U)−1 =Q2. Therefore, 1 0 ... 0 0 D=. , . Q2 .   0      and D is the diagonal matrix with elements 1,D ,D2,...,Dd−1 on the main 1 D1 Dd−2 diagonal (numbers D , j = 1,...,d−1, were defined in (4.2)). Let F be the j 10