A remark about Mahlers conjecture and the maximum value of box splines

A REMARK ABOUT MAHLER’S CONJECTURE AND THE MAXIMUM V ALUE OF BO X SPLINES ZHIQIANG XU Abstract. In this paper , we recast a specia l case of Mahler ’c conjecture by the maximum v alue of b ox splines. This is the cas e of po lytop es with a t mos t 2 n + 2 facets. An asymptotic form ula for univ ariate box splines is giv en. Based o n the formula, Mahler’s conjecture is prov ed in this c ase provided n is big eno ugh. 1. introduction Let K be a symmetric con ve x b o dy in R n , and let K ∗ b e its p olar { x : | h y , x i | ≤ 1 fo r a ll y ∈ K } . An old conjecture o f Mahler is (1) v ol( K ) · vol( K ∗ ) ≥ 4 n n ! . W e note that an n -dimensional par a llelepiped, whic h has 2 n facets, giv es equalit y in (1). So, t he first non-trivial case of Mahler’s conjecture is the symmetric conv ex b o dy K with 2 n + 2 facets. Suc h K can b e realized, up to a ffine inv ariance, as a one-co dimensional section o f an ( n + 1)-dimensional cub e. This case has b een raised as a separate problem b y Ball [1]. Ball also shows an in teresting relation b et w een the sp ecial case of Mahler’s conjecture and solutions o f a scaling equation. F or each r ∈ R , w e set (2) ϕ A ( r ) := v ol ( ( H + rA ) ∩ Q n ) where Q n := [ − 1 2 , 1 2 ] n , A := ( a 1 , . . . , a n ) is an unit vec to r and H := h A i ⊥ . Throughout this paper, without lo ss of generalit y , w e supp ose 0 < a 1 ≤ a 2 ≤ · · · ≤ a n . In particular, Ball sho ws that the sp ecial case of Mahler’s conjecture is equiv alen t to (3) ϕ A (0) · E      n X k =1 a k ε k      ! ≥ 1 , where ε k is a sequence of m utually independen t random v ariables with distribution P { 1 } = P {− 1 } = 1 / 2 . In [7], the authors prov e (3 ) fo r n ≤ 8 by reducing the problem to a searc h o v er a finite set of p olytop es for eac h fixed dimension. 1 2 ZHIQIAN G XU An in teresting observ ation is that ϕ A ( · ) is a b ox spline, a p opular to ol in appro ximation theory . Therefore, w e can recast the sp ecial case of Mahler’s conjecture b y the maxim um v alue o f b o x splines. Using the saddle p oin t appro ximation in statistics, w e giv e an asymptotic form ula of ϕ A ( · ). Based on the asymptotic form ula, w e obtain the follo wing result: Theorem 1. Supp ose that ther e is a c onstant c 0 so that a n /a 1 < c 0 for any n . Th en ther e exis ts a p ositive inte ger N 0 ( c 0 ) s o that ϕ A (0) · E      n X k =1 a k ε k      ! ≥ 1 when n ≥ N 0 ( c 0 ) . This pap er is organized as follows. In Section 2, w e in tro duce b ox splines and sho w the relation b et w een b ox splines and the special case of Mahler’s conjecture. In Section 3, w e use the saddle p oin t a pproxi- mation to giv e an asymptotic f orm ula of univ ariate b o x splines. Section 4 presen ts the pro of of Theorem 1. 2. B o x splines Supp ose that M is a s × n matrix. The b ox spline B ( ·| M ) a sso ciated with M is the distribution giv en by the rule [2, 3] (4) Z R s B ( x | M ) φ ( x ) d x = Z [0 , 1) n φ ( M u ) du, φ ∈ D ( R s ) . By taking φ = exp( − iζ · ) in (4), w e obta in the F ourier transform of B ( ·| M ) as (5) b B ( ζ | M ) = n Y j = 1 1 − exp( − iζ T m j ) iζ T m j , ζ ∈ C s . The follo wing form ula sho ws the relation b etw een b o x splines and the v olume of the section o f unit cub e (see [4], pag e 2): (6) B ( x | M ) = v ol n ( P ∩ [0 , 1) n ) p | det( M M T ) | , where P := { y : M y = x, y ∈ R n + } . Set A := ( a 1 , . . . , a n ). Recall that P j a 2 j = 1 and 0 < a 1 ≤ a 2 ≤ · · · ≤ a n . Then com bining (2) and ( 6) w e ha v e that ϕ A ( · ) = B ( · + ( a 1 + · · · + a n ) / 2 | A ) . MAHLER’S CONJECTURE AND BOX S PLINES 3 Also, noting B ( ·| A ) reac hing the maxim um v alue at ( a 1 + a 2 + · · · + a n ) / 2, (3) is equiv alen t to max x B ( x | A ) ≥ 1 E ( | P n k =1 a k ε k | ) . 3. An asymptotic formu la of univ aria te box s plines In this section, w e shall presen t an asymptotic fo r mula of B ( ·| A ). In [8], Unser et. al. prov ed that B ( ·| A ) tends to the Gaussian function as n increase pro vided a 1 = a 2 = · · · = a n . Here, using the saddle p oin t appro ximation in statistics, for the general matrix A , we can sho w the b ox spline B ( ·| A ) a lso conv erges t o the Gaussian function as n increases: Theorem 2. lim n →∞ B ( x | A ) = p 6 /π exp( − 6( x − X j a j / 2) 2 ) , wher e the limit may b e taken p ointwise or in L p ( R ) , p ∈ [2 , ∞ ) . Pr o of. The saddle p oin t approximation of B ( ·| A ) is (see Theorem 6.1 and 6 .2 in [5 ]) (7) 1 (2 π ) 1 / 2 | K ′′ ( s 0 ) | 1 / 2 exp( K ( s 0 ) − s 0 x ) . Here, K ( s ) := ln Y i exp( a i s ) − 1 a i s and s 0 satisfies K ′ ( s 0 ) = x . W e can consider s 0 as a function of x . So, (8) K ′ ( s 0 ) − x = 0 defines an implicit relationship b et w een s 0 and x . Noting K ′ (0) = ( a 1 + · · · + a n ) / 2, the equation (8) implies that s 0 (( a 1 + · · · + a n ) / 2) = 0. Also, b y (8), w e ha v e (9) K ′′ ( s 0 ) s ′ 0 − 1 = 0 , whic h implies that s ′ 0 (( a 1 + · · · + a n ) / 2) = 1 /K ′′ (0) = 12 / X j a 2 j = 12 . Using the similar metho d, we hav e s ′′ 0 (( a 1 + · · · + a n ) / 2) = 0 , 4 ZHIQIAN G XU and s ′′′ 0 (( a 1 + · · · + a n ) / 2) = 864 n X j = 1 a 4 j ! / 5 = O (1 /n ) . Using T a ylor expansion at ( a 1 + · · · + a n ) / 2, one has (10) s 0 ( x ) = 12 · ( x − ( a 1 + · · · + a n ) / 2) + O (1 /n ) . Also, b y (9), w e ha v e K ′′ ( s 0 ) = 1 /s ′ 0 = 1 / 12 + O (1 /n ) . Com bining T a ylor expansion o f K ( · ) at 0 and (10 ) , one has K ( s 0 ) = 6( X j a j )( x − ( a 1 + · · · + a n ) / 2)+6( x − ( a 1 + · · · + a n ) / 2) 2 + O (1 /n ) . By (10), w e hav e s 0 · x = 12( x − ( a 1 + · · · + a n ) / 2) x + O (1 / n ) = 12( x − ( a 1 + · · · + a n ) / 2) 2 + 6( X j a j )( x − ( a 1 + · · · + a n ) / 2) + O (1 /n ) . F rom (7), the saddle p oin t appro ximation of B ( x | A ) is p 6 /π exp( − 6( x − X j a j / 2) 2 ) + O (1 /n ) . The prop erties of t he saddle p oin t appro ximation imply this theorem.  4. p ro of of the main re sul t T o pro v e the main result, w e firstly in tro duce a lemma. Lemma 3. Put F ( s ) := 2 π Z ∞ 0 (1 −   cos( t/ √ s )   s ) t − 2 dt, s > 0 . Then E      n X k =1 a k ε k      ! ≥ F ( a − 2 n ) . Pr o of. By Lemma 1 .3 in [6], we hav e E | n X k =1 a k ε k | ! ≥ n X k =1 a 2 k F ( a − 2 k ) . MAHLER’S CONJECTURE AND BOX S PLINES 5 Since F is an increasing function (Lemma 1.4 [6]), w e ha v e E | n X k =1 a k ε k | ! ≥ n X k =1 a 2 k F ( a − 2 k ) ≥ F ( a − 2 n ) .  Pr o of of The or em 1. T o pro v e the theorem, we only need to pro v e that there exists a p ositiv e in teger N 0 so that max x B ( x | A ) ≥ 1 E ( | P n k =1 a k ε k | ) when n ≥ N 0 . Theorem 2 implies that (11) lim n →∞ max x B ( x | A ) = p 6 /π . Since P j a 2 j = 1 and a n /a 1 < c 0 , w e hav e lim n →∞ 1 /a n = ∞ . By [6], w e ha v e lim s →∞ F ( s ) = p 2 /π . W e c ho ose ε 0 so that 0 < ε 0 < p 6 /π − p π / 2 2 . By Lemma 3 , there is a p ositive integer N 1 so that (12) 1 E ( | P n k =1 a k ε k | ) ≤ 1 F ( a − 2 n ) ≤ p π / 2 + ε 0 pro vided n ≥ N 1 . The equation (11) implies that there is a p ositive in teger N 2 ( c 0 ) so tha t (13) max x B ( x | A ) ≥ p 6 /π − ε 0 pro vided n ≥ N 2 ( c 0 ). W e set N 0 ( c 0 ) := max { N 1 , N 2 ( c 0 ) } . Not ing t ha t p 6 /π − ε 0 ≥ p π / 2 + ε 0 , com bining (12) and (13), w e ha ve max B ( x | A ) ≥ 1 E ( | P n k =1 a k ε k | ) when n ≥ N 0 ( c 0 ).  Ac knowled gmen ts. Zhiqiang Xu w as supp orted b y National Natural Science F oundation of China (10871196 ) . 6 ZHIQIAN G XU Reference s 1. K. Ball, Mahler’s conjecture a nd wav elets, Discrete and Computational Ge- ometry , V ol. 13, No.1(199 5). 2. C. de Bo or and R. Devore, Approximation by smo oth m ultiv ariate s plines, T ra ns. Amer. Math. So c., 276 (1983 ) 775- 788. 3. C. de Boo r and K. H¨ ollig, B-splines fro m p a rallelepip eds, J. Anal. Math, 42(198 2/83)99 -115. 4. C. de Bo or, K. H¨ ollig and S. Riemenschneider, Box Splines, Springe r-V er lag, New Y ork , 1993 . 5. Daniels, H. E., Saddlep oint approximations in statistics , Annals Math. Statist., 25(195 4) 6 31-65 0 6. U. Haa gerup, The b est co nstants in the Khintc hine ineq ualit y , Studia Math- ematica, T. LXX. (1982). 7. M. A. Lop ez a nd S. Reisner, A sp ecia l case of Ma hler’s conjecture, Discre te Comp. Geo m. 20:163 -177(19 98). 8. M. Unser, A. Aldroubi a nd M. Eden, On the asymptotic conv er gence o f B- spline wav elets to Ga bo r functions, IE EE T ransactio ns on informations theory , V ol.38 , No.2, 19 92. Authors’ addresses: Zhiqiang Xu, LSEC, Inst. Comp. Math., Academ y of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China, xuzq@lsec.c c.ac.cn