CC − transfinite diameter N. Levenberg and F. WielonskyMarch 27, 2020
Abstract
We give a general formula for the C − transfinite diameter δ C ( K ) of a compactset K ⊂ C which is a product of univariate compacta where C ⊂ ( R + ) is a convexbody. Along the way we prove a Rumely type formula relating δ C ( K ) and the C − Robin function ρ V C,K of the C − extremal plurisubharmonic function V C,K for C ⊂ ( R + ) a triangle T a,b with vertices (0 , , ( b, , (0 , a ). Finally, we show howthe definition of δ C ( K ) can be extended to include many nonconvex bodies C ⊂ R d for d − circled sets K ⊂ C d , and we prove an integral formula for δ C ( K ) which weuse to compute a formula for the C − transfinite diameter of the Euclidean unit ball B ⊂ C . In the recently developed pluripotential theory associated to a convex body C ⊂ ( R + ) d (cf., [1]), notions of C − extremal plurisubharmonic (psh) function V C,K and C − transfinitediameter δ C ( K ) of a compact set K ⊂ C d generalize the corresponding notions in thestandard setting. Their definitions are recalled in the next section, and we include abrief discussion of Ma‘u’s recent work [12] on C − transfinite diameter. We also recall thenotion of C − Robin function ρ V C,K associated to V C,K as defined in [9] for C ⊂ ( R + ) atriangle T a,b with vertices (0 , , ( b, , (0 , a ). The C − Robin function describes the preciseasymptotic behavior of V C,K ; i.e., the behavior of V C,K ( z ) for | z | large.In classical pluripotential theory, which corresponds to the special case where C isthe standard unit simplex Σ ⊂ ( R + ) d , it is very difficult to find explicit formulas forextremal psh functions V K (and hence their Robin functions) or to find precise values oftransfinite diameters δ d ( K ) for K ⊂ C d . In 1962, Schiffer and Siciak [13] proved that if K = E × · · · × E d ⊂ C d is a product of planar compact sets E j , then δ d ( K ) = (cid:81) dj =1 D ( E j )where D ( E j ) is the univariate transfinite diameter of E j . Their proof used an intertwiningof univariate Leja sequences for the sets E j . Then in 1999, Bloom and Calvi [4] proved amore general result: if K = E × F where E ⊂ C m and F ⊂ C n , then δ n + m ( K ) = Ä δ m ( E ) m · δ n ( F ) n ä m + n . (1.1)Their proof used orthogonal polynomials associated to certain measures, called Bernstein-Markov measures, on K . In 2005, Calvi and Phung Van Manh [6] recovered the Bloom-Calvi result (1.1) by generalizing the Schiffer-Siciak method in introducing “block” Lejasequences for the component sets. 1 a r X i v : . [ m a t h . C V ] M a r n [10], Rumely gave a remarkable formula relating transfinite diameter and Robinfunction in this classical setting. Using this formula, Blocki, Edigarian and Siciak [3] gavea very short proof of the general product formula (1.1). In section 3, based on results in [1]and [9], we prove a Rumely type formula relating δ C ( K ) and ρ V C,K for C = T a,b ⊂ ( R + ) and we use this in section 4 to prove a formula for δ C ( K ) when K = E × F is a productof univariate compacta. We modify the Bloom-Calvi proof using orthogonal polynomialsin section 5 to give a product formula for the C − transfinite diameter when C is a generalconvex body in ( R + ) . In particular, for such C which are symmetric with respect to theline y = x , we obtain the striking result that the C − transfinite diameter of K = E × F is the same for these C . Finally, in section 6, we show how the C − transfinite diameter δ C ( K ) can be extended to include many nonconvex bodies C ⊂ R d for d − circled sets K ⊂ C d , and we exhibit an integral formula for δ C ( K ). We use this to directly compute aformula for δ C p ( B ) for the Euclidean unit ball B ⊂ C for a natural one-parameter familyof symmetric C = C p (section 6) which explicitly yields different values for different p . C − transfinite diameter and C − Robin function
Let C be a convex body in ( R + ) d . We assume throughout that (cid:15) Σ ⊂ C ⊂ δ Σ for some δ > (cid:15) > { ( x , ..., x d ) ∈ R d : 0 ≤ x i ≤ , d (cid:88) j =1 x i ≤ } . We setPoly( nC ) = { p ( z ) = (cid:88) J ∈ nC ∩ N d c J z J = (cid:88) J ∈ nC ∩ N d c J z j · · · z j d d , c J ∈ C } , n = 1 , , . . . and for a nonconstant polynomial p we definedeg C ( p ) = min { n ∈ N : p ∈ Poly( nC ) } . Next, we define the logarithmic indicator function H C ( z ) := sup J ∈ C log | z J | := sup ( j ,...,j d ) ∈ C log Ä | z | j · · · | z d | j d ä in order to define L C = L C ( C d ) := { u ∈ P SH ( C d ) : u ( z ) − H C ( z ) = O (1) , | z | → ∞} , and L + C = L + C ( C d ) = { u ∈ L C ( C d ) : u ( z ) ≥ H C ( z ) + C u } where P SH ( C d ) denotes the class of plurisubharmonic functions on C d . In particular,if p ∈ Poly( nC ) then u ( z ) := 1deg C ( p ) log | p ( z ) | ∈ L C . L := L Σ , L + := L +Σ when C = Σ. The C -extremal function of a compact set K ⊂ C d is defined as theuppersemicontinuous (usc) regularization V ∗ C,K ( z ) := lim sup ζ → z V C,K ( ζ ) of V C,K ( z ) := sup { u ( z ) : u ∈ L C , u ≤ K } . If C = Σ, we simply write V K := V Σ ,K . As in this classical setting, V ∗ C,K ≡ + ∞ if and onlyif K is pluripolar; and when this is not the case, the complex Monge-Amp`ere measure( dd c V ∗ C,K ) d is supported in K . We call K regular if V K = V ∗ K ; i.e., V K is continuous.This is equivalent to V C,K being continuous for any C . Our definition of dd c is such that( dd c log + max[ | z | , ..., | z d | ]) d is a probability measure.We recall the definition of C − transfinite diameter δ C ( K ) of a compact set K ⊂ C d .Letting d n be the dimension of Poly( nC ), we havePoly( nC ) = span { e , ..., e d n } where { e j ( z ) := z α ( j ) = z α ( j )1 · · · z α d ( j ) d } j =1 ,...,d n are the standard basis monomials inPoly( nC ) in any order. For points ζ , ..., ζ d n ∈ C d , let V DM ( ζ , ..., ζ d n ) := det[ e i ( ζ j )] i,j =1 ,...,d n = det e ( ζ ) e ( ζ ) . . . e ( ζ d n )... ... . . . ... e d n ( ζ ) e d n ( ζ ) . . . e d n ( ζ d n ) and for a compact subset K ⊂ C d let V n = V n ( K ) := max ζ ,...,ζ dn ∈ K | V DM ( ζ , ..., ζ d n ) | . (2.2)Then δ C ( K ) := lim sup n →∞ V /l n n (2.3)is the C − transfinite diameter of K where l n := (cid:80) d n j =1 deg( e j ).The existence of the limit is not obvious. In this generality it was proved in [1]. In theclassical ( C = Σ) case, Zaharjuta [14] verified the existence of the limit by introducingdirectional Chebyshev constants τ ( K, θ ) and proving δ Σ ( K ) = exp Ä | σ | (cid:90) σ log τ ( K, θ ) d | σ | ( θ ) ä where σ := { ( x , ..., x d ) ∈ R d : 0 ≤ x i ≤ , (cid:80) dj =1 x i = 1 } is the extreme “face” of Σ; σ = { ( x , ..., x d ) ∈ R d : 0 < x i < , (cid:80) dj =1 x i = 1 } ; and | σ | is the ( d − − dimensionalmeasure of σ . We will utilize results from [12] where a Zaharjuta-type proof of theexistence of the limit in the general C − setting is given. There it is shown that δ C ( K ) = î exp Ä vol ( C ) (cid:90) C o log τ C ( K, θ ) dm ( θ ) äó /A C (2.4)where the directional Chebyshev constants τ C ( K, θ ) and the integration in the formulaare over the interior C o of the entire d − dimensional convex body C and A C is a positiveconstant depending only on C and d (defined in (2.9)).3priori, in the definition of τ C ( K, θ ) the standard grlex (graded lexicographic) ordering ≺ on N d (i.e., on the monomials in C d ) was used. This was required to obtain thesubmultiplicativity of the “monic” polynomial classes M k ( α ) := { p ∈ Poly( kC ) : p ( z ) = z α + (cid:88) β ∈ kC ∩ N d , β ≺ α c β z β } (2.5)for α ∈ kC ∩ N d ; i.e., M k ( α ) · M k ( α ) ⊂ M k + k ( α + α ). Defining Chebyshev constants T k ( K, α ) := inf {(cid:107) p (cid:107) K : p ∈ M k ( α ) } /k , (2.6)for θ ∈ C o , this submultiplicativity allows one to verify existence of the limit τ C ( K, θ ) := lim k →∞ , α/k → θ T k ( K, α ) (2.7)as well as convexity of the function θ → ln τ C ( K, θ ) on C o .In the proof that lim n →∞ V /l n n exists in [12], it is shown thatlim n →∞ V /nd n n = lim n →∞ Ä d n (cid:89) j =1 T n ( K, α ( j )) n ä /nd n . (2.8)The asymptotic relation between nd n and l n is thatlim n →∞ l n nd n = A C := 1vol( C ) · (cid:90)(cid:90) C ( x + · · · x d ) dx · · · dx d =: M C / vol( C ) . (2.9)The following propositions will be useful in the sequel. Proposition 2.1.
For t > , δ tC ( K ) = δ C ( K ) . Proof.
We first observe that if t ∈ N , since the limit in (2.3) exists, δ C ( K ) = lim n →∞ V /l n n = lim n →∞ V /l tn tn = δ tC ( K ) . Similarly, if t ∈ Q we have δ tC ( K ) = δ C ( K ). To verify the result for t ∈ R , we proceedas follows. If t < t < t , from the definitions of M k ( α ) , T k ( K, α ) and τ C ( K, θ ), we havethe following:1. for θ ∈ t C o , τ t C ( K, θ ) ≥ τ tC ( K, θ ); and2. for θ ∈ tC o , τ tC ( K, θ ) ≥ τ tC ( K, θ ).Taking a sequence { t ,j } ⊂ Q with t ,j ↑ t and a sequence { t ,j } ⊂ Q with t ,j ↓ t , usingthe above inequalities together with (2.4) and (2.9),lim j →∞ δ t ,j C ( K ) = lim j →∞ δ t ,j C ( K ) = δ tC ( K ) . We can use the Hausdorff metric on the family of our convex bodies C satisfying (2.1)considered as compact sets in R d . Using similar ideas from the previous proof, we verifythe next result. 4 roposition 2.2. Given K ⊂ C d , the mapping C → δ C ( K ) is continuous.Proof. Taking a sequence { C j } of convex bodies satisfying (2.1) converging to C in theHausdorff metric, we can find (cid:15) j → − (cid:15) j ) C ⊂ C j ⊂ (1 + (cid:15) j ) C, j = 1 , , ... As in the proof of Proposition 2.1, we have1. for θ ∈ (1 − (cid:15) j ) C o , τ (1 − (cid:15) j ) C ( K, θ ) ≥ τ C j ( K, θ ); and2. for θ ∈ C oj , τ C j ( K, θ ) ≥ τ (1+ (cid:15) j ) C ( K, θ ).Since (cid:15) j → C j ) → vol( C ) and M C j → M C , using the above inequalitiestogether with (2.4) and (2.9), we find lim j →∞ δ C j ( K ) = δ C ( K ).For most of the subsequent sections, we work in C . First, recall the definition of theRobin function ρ u associated to u ∈ L ( C ): ρ u ( z ) := lim sup | λ |→∞ [ u ( λz ) − log | λ | ] . For z = ( z , z ) (cid:54) = (0 ,
0) we define ρ u ( z ) := lim sup | λ |→∞ [ u ( λz ) − log | λz | ] = ρ u ( z ) − log | z | so that ρ u ( tz ) = ρ u ( z ) for t ∈ C \ { } . Here | z | = | z | + | z | . We can consider ρ u as afunction on P = P \ C where to p = ( p , p ) with | p | = 1 we associate the point wherethe complex line λ → λp hits P .For a special class of convex bodies, there is a generalization of the notion of Robinfunction. Following [9], if we let C be the triangle T a,b with vertices (0 , , ( b, , (0 , a )where a, b are relatively prime positive integers, we have the following:1. H C ( z , z ) = max[log + | z | b , log + | z | a ] (note H C = 0 on the closure of the unitpolydisk P := { ( z , z ) : | z | , | z | < } ), and, indeed, H C = V C,P = V C,T where T := { ( z , z ) : | z | , | z | = 1 } ;2. defining λ ◦ ( z , z ) := ( λ a z , λ b z ), we have H C ( λ ◦ ( z , z )) = H C ( z , z ) + ab log | λ | for ( z , z ) ∈ C \ P and | λ | ≥ Definition 2.3.
Given u ∈ L C , we define the C − Robin function of u : ρ u ( z , z ) := lim sup | λ |→∞ [ u ( λ ◦ ( z , z )) − ab log | λ | ]for ( z , z ) ∈ C . 5pplying the transformation formula Theorem 4.1 of [9] in the case where d = 2; C is our triangle with vertices (0 , , ( b, , (0 , a ); C (cid:48) = ab Σ; and we consider the properpolynomial mapping F ( z , z ) = ( z a , z b ) , we obtain abV F − ( K ) ( z , z ) = V C,K ( z a , z b )so that abρ V F − ( K ) ( z , z ) = lim sup | λ |→∞ [ V C,K ( λ a z a , λ b z b ) − ab log | λ | ]= lim sup | λ |→∞ [ V C,K ( λ ◦ ( z a , z b )) − ab log | λ | ]= ρ V C,K ( z a , z b ) = ρ V C,K ( F ( z , z )) . More generally, letting ζ = ( ζ , ζ ) = F ( z ) = F ( z , z ) = ( z a , z b ) , for u ∈ L C , we have (cid:101) u ( z ) := u ( F ( z , z )) = u ( ζ ) ∈ abL and (2.10) ρ u ( ζ ) = ρ u ( F ( z , z )) = abρ (cid:101) u / ab ( z ) (2.11)where ρ (cid:101) u / ab is the standard Robin function of (cid:101) u/ab ∈ L . Note that if u ∈ L + C then (cid:101) u ∈ abL + . We apply these results in the next section. C − Rumely formula for C = T a,b In this section, we let C = T a,b . We begin with some integral formulas associated to func-tions in L + ( C ). The integral formula Theorem 5.5 of [2] in this setting is the following. Theorem 3.1. (Bedford-Taylor)
Let u, v, w ∈ L + ( C ) . Then (cid:90) C ( udd c v − vdd c u ) ∧ dd c w = (cid:90) P ( ρ u − ρ v )( dd c ρ w + ω ) where ω is the standard K¨ahler form on P . Next, following the arguments in [7], we get a symmetrized integral formula involvingRobin functions ρ u , ρ v for u, v ∈ L + ( C ) and their projectivized versions ρ u , ρ v : (cid:90) P ( ρ u − ρ v ) î ( dd c ρ u + ω ) + ( dd c ρ v + ω )] (3.1)= (cid:90) C ρ u (1 , t ) dd c ρ u (1 , t ) + ρ u (0 , − [ (cid:90) C ρ v (1 , t ) dd c ρ v (1 , t ) + ρ v (0 , . From (2.10), if u, v, w ∈ L + C , ab (cid:90) C ( udd c v − vdd c u ) ∧ dd c w = (cid:90) C ( (cid:101) udd c (cid:101) v − (cid:101) vdd c (cid:101) u ) ∧ dd c (cid:102) w.
6e apply Theorem 3.1 to the right-hand-side, multiplying by factors of ab since (cid:101) u, (cid:101) v, (cid:102) w ∈ abL + , to obtain, with the aid of (2.11), the desired integral formula (cf., (6.3) in [9]): (cid:90) C ( udd c v − vdd c u ) ∧ dd c w = ( ab ) (cid:90) P ( ρ (cid:101) u / ab − ρ (cid:101) v / ab )( dd c ρ (cid:101) w / ab + ω ) . (3.2)Next, for u, v ∈ L + C , we define the mutual energy E ( u, v ) := (cid:90) C ( u − v )[( dd c u ) + dd c u ∧ dd c v + ( dd c v ) ] . (cf., (3.1) in [1]). We connect this notion with C − transfinite diameter by recalling thefollowing formula from [1]. Theorem 3.2.
Let K ⊂ C be compact and nonpluripolar. Then log δ C ( K ) = − c E ( V ∗ C,K , H C ) where c = 3! M C with M C := (cid:82)(cid:82) C ( x + y ) dxdy . Remark 3.3.
This formula is actually valid in C d for d > C ⊂ ( R + ) d satisfying (2.1) with the appropriate definitions of E and c .Our goal in this section is to rewrite E ( V ∗ C,K , H C ) using the integral formulas in orderto get a formula relating δ C ( K ) and ρ (cid:101) V C , K / ab more in the spirit of Proposition 3.1 in [7].This will be used in the next section to prove a formula for the C − transfinite diameter δ C ( K ) of a product set K = E × F . Proposition 3.4.
We have E ( V ∗ C,K , H C ) = ( ab ) [ (cid:90) C ρ (cid:101) V C , K / ab (1 , t ) dd c ρ (cid:101) V C , K / ab (1 , t ) − ρ (cid:101) V C , K / ab (0 , . Hence from Theorem 3.2 − M C log δ C ( K ) = ( ab ) [ (cid:90) C ρ (cid:101) V C , K / ab (1 , t ) dd c ρ (cid:101) V C , K / ab (1 , t ) − ρ (cid:101) V C , K / ab (0 , . (3.3) Proof.
Applying the formula (3.2) with w = u and w = v and adding, we obtain (cid:90) C ( udd c v − vdd c u ) ∧ dd c ( u + v ) = ( ab ) (cid:90) P ( ρ (cid:101) u / ab − ρ (cid:101) v / ab )[( dd c ρ (cid:101) u / ab + ω ) + ( dd c ρ (cid:101) v / ab + ω )] . We claim from the definition of E ( u, v ), it follows that E ( u, v ) = (cid:90) C [ u ( dd c v ) − v ( dd c u ) ]+( ab ) (cid:90) P ( ρ (cid:101) u / ab − ρ (cid:101) v / ab )[( dd c ρ (cid:101) u / ab + ω )+( dd c ρ (cid:101) v / ab + ω )] . (3.4)To see this, using the previous formula it suffices to show E ( u, v ) − (cid:90) C u ( dd c u ) + (cid:90) C v ( dd c v ) = (cid:90) C ( udd c v − vdd c u ) ∧ dd c ( u + v ) . In verifying this, all integrals are over C . We write E ( u, v ) = (cid:90) ( u − v )[( dd c u ) + ( dd c v ∧ dd c ( u + v )]7 (cid:90) u ( dd c u ) − (cid:90) v ( dd c u ) + (cid:90) ( u − v ) dd c v ∧ dd c ( u + v )= (cid:90) u ( dd c u ) − (cid:90) v ( dd c v ) + (cid:90) ( u − v ) dd c v ∧ dd c ( u + v ) + (cid:90) v [( dd c v ) − ( dd c u ) ] . We finish this proof by working with the sum of the last two integrals: (cid:90) ( u − v ) dd c v ∧ dd c ( u + v ) + (cid:90) v [( dd c v ) − ( dd c u ) ]= (cid:90) ( u − v ) dd c v ∧ dd c ( u + v ) + (cid:90) v [ dd c ( v − u ) ∧ dd c ( u + v )]= (cid:90) ( udd c v − vdd c u ) ∧ dd c ( u + v )as desired.Letting u = V C,K and v = V C,K in (3.4) where K , K are regular compact sets in C , E ( V C,K , V C,K ) = ( ab ) (cid:90) P ( ρ (cid:101) V C , K1 / ab − ρ (cid:101) V C , K2 / ab )[( dd c ρ (cid:101) V C , K1 / ab + ω )+( dd c ρ (cid:101) V C , K2 / ab + ω )] . In particular, since H C = V C,P = V C,T where T is the unit torus in C , E ( V C,K , H C ) = ( ab ) (cid:90) P ( ρ (cid:101) V C , K1 / ab − ρ (cid:101) H C / ab )[( dd c ρ (cid:101) V C , K1 / ab + ω ) + ( dd c ρ (cid:101) H C / ab + ω )] . The result will follow from (3.1) once we verify (cid:90) C ρ (cid:101) H C / ab (1 , t ) dd c ρ (cid:101) H C / ab (1 , t ) + ρ (cid:101) H C / ab (0 ,
1) = 0 . (3.5)To verify (3.5), we begin by observing that since H C ( z , z ) = max Ä log + | z | b , log + | z | a ä , (cid:102) H C ( z , z ) := H C ( z a , z b ) , for ( z , z ) ∈ C \ ( P ) o , ρ H C ( z a , z b ) = H C ( z a , z b ) = abρ (cid:101) H C / ab ( z , z ) . In particular, ρ (cid:101) H C / ab (0 ,
1) = 1 ab H C (0 ,
1) = 0 and ρ (cid:101) H C / ab (1 , t ) = 1 ab H C (1 , t b ) = 1 ab max (0 , ab log | t | ) . Thus dd c ρ (cid:101) H C / ab (1 , t ) is supported on | t | = 1 where it is (normalized) arclength measure.On this set, we have ρ (cid:101) H C / ab (1 , t ) = 0 and (3.5) follows.8 Product formula for C a triangle We first use (3.3) to prove a formula for the C − transfinite diameter of a product set when C is a triangle T a,b with vertices (0 , , ( b, , (0 , a ). Then in the next section we give a(conceptually) simpler proof that is valid for general convex bodies. Theorem 4.1.
Let K = E × F where E, F ⊂ C are compact. Then for C = T a,b , − log δ C ( K ) = aba + b · Ç − log D ( E ) a + − log D ( F ) b å ; i.e., δ C ( K ) = D ( E ) b/ ( a + b ) D ( F ) a/ ( a + b ) where D ( E ) , D ( F ) are the univariate transfinite diameters of E, F .Proof.
We first assume a, b are positive integers and use Proposition 3.4. To this end, wecompute ρ (cid:101) V C , K / ab for K = E × F . We can assume E, F are regular compact sets in C and we let ρ E = − log D ( E ) and ρ F = − log D ( F ) be the Robin constants of these sets.From Proposition 2.4 of [5], V C,K ( z , z ) = max ( bg E ( z ) , ag F ( z ))where g E , g F are the Green functions for E, F . Note that ρ E = lim | z |→∞ [ g E ( z ) − log | z | ] and ρ F = lim | z |→∞ [ g F ( z ) − log | z | ] . Thus from Definition 2.3 ρ V C,K ( z , z ) = lim sup | λ |→∞ [max[ bg E ( λ a z ) , ag F ( λ b z )] − ab log | λ | ]= max[ b ( ρ E + log | z | ) , a ( ρ F + log | z | )]so that ρ (cid:101) V C , K / ab ( z , z ) = 1 ab ρ V C,K ( z a , z b ) = max[ 1 a ρ E + log | z | , b ρ F + log | z | ] . Hence ρ (cid:101) V C , K / ab (1 , t ) = max Ç a ρ E , b ρ F + log | t | å so that dd c ρ (cid:101) V C , K / ab (1 , t ) is normalized arclength meaure on a circle where the value ofthe function ρ (cid:101) V C , K / ab (1 , t ) = a ρ E . Finally, ρ (cid:101) V C , K / ab (0 ,
1) = b ρ F and the result when a, b are positive integers follows from Proposition 3.4 since( ab ) [ (cid:90) C ρ (cid:101) V C , K / ab (1 , t ) dd c ρ (cid:101) V C , K / ab (1 , t ) − ρ (cid:101) V C , K / ab (0 , ab ) Ç a ρ E + 1 b ρ F å and a calculation shows that M C = ( ab/ a + b ) so that 3! M C = ( ab )( a + b ). If a, b ∈ Q ,the result follows from Proposition 2.1; finally, the general case when a, b ∈ R followsfrom Proposition 2.2. 9 Product formula for general C In this section, we give an alternate proof of Theorem 4.1 which is applicable in a muchmore general setting. We assume that C is a convex body satisfying (2.1) which is a lowerset : whenever ( j , j ) ∈ nC ∩ N we have ( k , k ) ∈ nC ∩ N for all k l ≤ j l , l = 1 ,
2. Forexample, the triangles T a,b are lower sets. This proof is modeled on that of Bloom-Calviin [4]. As in the previous section, we take K = E × F where E, F are compact sets in C . Let µ E , µ F be Bernstein-Markov measures for
E, F : recall ν is a Bernstein-Markovmeasure for E if for any (cid:15) >
0, there exists a constant c (cid:15) so that (cid:107) p n (cid:107) K ≤ c (cid:15) (1 + (cid:15) ) n (cid:107) p n (cid:107) L ( ν ) , n = 1 , , ... where p n is any polynomial of degree n . If E, F are regular, one can take, e.g., µ E and µ F to be the distributional Laplacians of the Green functions g E and g F . Let µ := µ E ⊗ µ F .Let { p j ( z ) } j =0 , , ,... be monic orthogonal polynomials for L ( µ E ) and let { q k ( z ) } k =0 , , ,... be monic orthogonal polynomials for L ( µ F ); then { p j ( z ) q k ( w ) } j,k =0 , , ,... are orthogonalin L ( µ ). Using the grlex ordering ≺ on N and the lower set property of C , it is easy tosee that each L ( µ ) − orthogonal polynomial p j ( z ) q k ( w ) is in a class M l ( α ) (recall (2.5))where α = ( j, k ) and l = deg C ( z j w k ). Here and below j, k are nonnegative integers.We want to use (2.8): the asymptotics of V n and (cid:81) d n j =1 T n ( K, α ( j )) n are the same; i.e.,the limits of their nd n − th roots coincide. If µ is a Bernstein-Markov measure on K , itfollows readily that one can replace the sup-norm minimizers T k ( K, α ) by L ( µ ) − normminimizers (cid:102) T k ( K, α ) := inf {(cid:107) p (cid:107) L ( µ ) : p ∈ M k ( α ) } /k . In our setting, for α = ( j, k ) the polynomial p j ( z ) q k ( w ) is the minimizer and (cid:107) p j q k (cid:107) L ( µ ) = (cid:107) p j (cid:107) L ( µ E ) · (cid:107) q k (cid:107) L ( µ F ) . Moreover, we know from the univariate theory thatlim j →∞ (cid:107) p j (cid:107) /jL ( µ E ) = D ( E ) and lim k →∞ (cid:107) q k (cid:107) /kL ( µ F ) = D ( F ) . For simplicity, we write p j := (cid:107) p j (cid:107) L ( µ E ) and q k := (cid:107) q k (cid:107) L ( µ F ) . In this notation, to utilize (2.8), we consider Ñ (cid:89) ( j,k ) ∈ nC p j q k é /nd n . We suppose that ( b,
0) and (0 , a ) are extreme points of C and that the outer face F C of C ; i.e., the portion of the topological boundary of C outside of the coordinate axes, canbe written both as a graph { ( x, f ( x )) : 0 ≤ x ≤ b } and as a graph { ( g ( y ) , y ) : 0 ≤ y ≤ a } . Theorem 5.1.
Let K = E × F where E, F are compact subsets of C . Then δ C ( K ) = D ( E ) A/ ( A + B ) · D ( F ) B/ ( A + B ) (5.1) where A = (cid:82) b uf ( u ) du and B = (cid:82) a ug ( u ) du . Hence for any convex body C with A = B we obtain δ C ( K ) = [ D ( E ) D ( F )] / . roof. The outer face F nC of nC can be written as { ( x, y ) : y = f n ( x ) := nf ( x/n ) , ≤ x ≤ nb } = { ( x, y ) : x = g n ( y ) := ng ( y/n ) , ≤ y ≤ na } . Then the product (cid:81) ( j,k ) ∈ nC p j q k ; i.e., the product over the integer lattice points in nC , isasymptotically given by p f n (1)1 p f n (2)2 · · · p f n ( nb ) nb q g n (1)1 q g n (2)2 · · · q g n ( na ) na = (cid:16) p f (1 /n )1 p f (2 /n )2 · · · p f ( nb/n ) nb q g (1 /n )1 q g (2 /n )2 · · · q g ( na/n ) na (cid:17) n . For simplicity in the calculation, we concentrate on the product p f (1 /n )1 p f (2 /n )2 · · · p f ( nb/n ) nb Then using the fact that p j (cid:16) D ( E ) j , p f (1 /n )1 p f (2 /n )2 · · · p f ( nb/n ) nb (cid:16) D ( E ) f (1 /n )+2 f (2 /n )+ ··· + nbf ( nb/n ) = D ( E ) n · (1 /n )[1 /nf (1 /n )+2 /nf (2 /n )+ ··· + nb/nf ( nb/n )] (cid:16) D ( E ) n (cid:82) b uf ( u ) du . Similarly, since q k (cid:16) D ( F ) k , we have q g (1 /n )1 q g (2 /n )2 · · · q g ( na/n ) na (cid:16) D ( F ) n (cid:82) a ug ( u ) du . Hence (cid:89) ( j,k ) ∈ nC p j q k (cid:16) D ( E ) n (cid:82) b uf ( u ) du · D ( F ) n (cid:82) a ug ( u ) du . Using (2.8) and (2.9), since nd n A C (cid:16) nn area( C ) · M C area( C ) = n M C , and M C = (cid:90)(cid:90) C xdydx + (cid:90)(cid:90) C ydxdy = (cid:90) b (cid:90) f ( x )0 xdydx + (cid:90) a (cid:90) g ( y )0 ydxdy = (cid:90) b xf ( x ) dx + (cid:90) a yg ( y ) dy = A + B, (5.1) follows. Remark 5.2.
Note that A = B occurs whenever a = b and f = g ; i.e., the convex bodyis symmetric about the line y = x . As special cases of this, we can take C = C p := { ( x, y ) : x, y ≥ , x p + y p ≤ } , ≤ p < ∞ (5.2)as well as C ∞ := { ( x, y ) : 0 ≤ x, y ≤ } . emark 5.3. We can use Theorem 5.1 to verify our result in Theorem 4.1 for C = T a,b .Here y = f ( x ) = a (1 − x/b ) , ≤ x ≤ b and x = g ( y ) = b (1 − y/a ) , ≤ y ≤ a . Then (cid:90) b xf ( x ) dx = ab (cid:90) a yg ( y ) dy = ba M T a,b = (cid:90) b (cid:90) a (1 − x/b )0 ( x + y ) dydx = ( ab/ a + b ) . Hence δ T a,b ( K ) = D ( E ) b/ ( a + b ) D ( F ) a/ ( a + b ) . Moreover, the calculations in Theorem 5.1 – and the resulting formula – are valid (andmuch easier) in a special case where F C cannot be written as a graph { ( x, f ( x )) : 0 ≤ x ≤ b } (nor as a graph { ( g ( y ) , y ) : 0 ≤ y ≤ a } ); namely, the rectangle C = R a,b withvertices (0 , , ( b, , (0 , a ) and ( b, a ). Here we take y = f ( x ) = a, ≤ x ≤ b and x = g ( y ) = b, ≤ y ≤ a and the calculations in the proof of Theorem 5.1 yield (cid:90) b xf ( x ) dx = ab (cid:90) a xyg ( y ) dy = ba M R a,b = (cid:90) b (cid:90) a ( x + y ) dydx = ( ab/ a + b ) . Hence, for this rectangle we recover the same product formula as for T a,b : δ R a,b ( K ) = D ( E ) b/ ( a + b ) D ( F ) a/ ( a + b ) . d − circled sets One might wonder, given Remark 5.2, whether we always have equality of δ C ( K ) for allconvex bodies C that are symmetric about the line y = x (e.g., C p for 1 ≤ p ≤ ∞ ), i.e.,for any compact set K , not just product sets. This is not the case as we will illustrate for B := { ( z , z ) : | z | + | z | ≤ } , the closed Euclidean unit ball in C . This is an example of a 2 − circled set. We say a set E ⊂ C d is d − circled if( z , ..., z d ) ∈ E implies ( e iβ z , ..., e iβ d z d ) ∈ E, for all real β , ..., β d . For a compact, d − circled set K , it is easy to see from the Cauchy estimates thatinf {(cid:107) p (cid:107) K : p ∈ M k ( α ) } = (cid:107) z α (cid:107) K where recall M k ( α ) := { p ∈ Poly( kC ) : p ( z ) = z α + (cid:88) β ∈ kC ∩ N d , β ≺ α c β z β } . C satisfying (2.1), and any θ = ( θ , ..., θ d ) ∈ C o , we have τ C ( K, θ ) := lim k →∞ , α/k → θ T k ( K, α ) = lim k →∞ , α/k → θ (cid:107) z α (cid:107) /kK = max z ∈ K | z | θ · · · | z d | θ d . Thus, for a given d − circled set K , if we can explicitly determine these values, we can use(2.4) to compute δ C ( K ).Indeed, an elementary calculation for K = B ⊂ C shows that τ C ( B , θ ) = Ç θ θ + θ å θ / Ç θ θ + θ å θ / . (6.1)It follows readily from (2.4) that δ C ( B ) = e − / .We next show that the main result in [12], specifically, equation (2.4) in our Section1, remains valid even for certain nonconvex sets C and all d − circled sets K . To this end,let C ⊂ ( R + ) d be the closure of an open, connected set satisfying (2.1). As examples, onecan take C p as in (5.2) for 0 < p <
1. Here, the definition ofPoly( nC ) = { p ( z ) = (cid:88) J ∈ nC ∩ N d c J z J , c J ∈ C } , n = 1 , , . . . makes sense; and we have Poly( nC ) = span { e , ..., e d n } where e j ( z ) := z α ( j ) are thestandard basis monomials in Poly( nC ) and d n is the dimension of Poly( nC ). Using thesame notation V DM ( ζ , ..., ζ d n ) := det[ e i ( ζ j )] i,j =1 ,...,d n as in the convex setting, for a compact and d − circled set K ⊂ C d , we have the samenotions of maximal Vandermonde V n = V n ( K ); C − transfinite diameter δ C ( K ); “monic”polynomial classes M k ( α ) and corresponding Chebyshev constants T k ( K, α ); and direc-tional Chebyshev constants τ C ( K, θ ) for θ ∈ C o as in (2.2), (2.3), (2.5), (2.6) and (2.7). Proposition 6.1.
For C ∈ ( R + ) d the closure of an open, connected set satisfying (2.1)and for any d − circled set K ⊂ C d , we have:1. for θ ∈ C o , τ C ( K, θ ) := lim k →∞ , α/k → θ T k ( K, α ) , i.e., the limit exists; and2. lim n →∞ V /nd n n exists and equals lim n →∞ [ (cid:81) d n j =1 T n ( K, α ( j )) n ] /nd n ;3. δ C ( K ) = î exp Ä vol ( C ) (cid:82) C o log τ C ( K, θ ) dm ( θ ) äó /A C where A C is a positive constantdefined in (2.9).Proof. Because inf {(cid:107) p (cid:107) K : p ∈ M k ( α ) } = (cid:107) z α (cid:107) K , all the arguments in Lemmas 4.4 and4.5 of [12] work to show d n (cid:89) j =1 T n ( K, α ( j )) n ≤ V n ≤ d n ! · d n (cid:89) j =1 T n ( K, α ( j )) n . The only other ingredients needed to complete the rest of the proof are simply to observethat even though the polynomial classes M k ( α ) are not submultiplicative, the monomials z α themselves are; i.e., z α z β = z α + β ∈ M k ( α + β ). This is all that is needed to show 1.;then the proof in [12] gives 2. and 3. 13rom the general formula τ C ( K, θ ) = max z ∈ K | z | θ · · · | z d | θ d for any d − circled set K ⊂ C d , δ C ( K ) = Ç exp Ç vol ( C ) (cid:90) C o log Ä max z ∈ K | z | θ · · · | z d | θ d ä dm ( θ ) åå /A C . Using (6.1), for K = B ⊂ C , δ C ( B ) = (cid:32) exp (cid:32) C ) (cid:90) C o log (cid:32) Ç θ θ + θ å θ / Ç θ θ + θ å θ / (cid:33) dm ( θ ) (cid:33)(cid:33) /A C . (6.2)Note that for C = C p this gives a formula for the C p -transfinite diameter of the ball B in C valid for all 0 < p ≤ ∞ . We return to this approach to computing C − transfinitediameter using directional Chebyshev constants in Proposition 6.5.We can also use orthogonal polynomials as in Section 5 to compute δ C ( B ) for general C as in Proposition 6.1; this we do next. Proposition 6.2.
For C as in Proposition 6.1, the C -transfinite diameter of the ball B in C is equal to δ C ( B ) = exp Ç I Ç I + I − I − log 2 π C ) åå , (6.3) where I = (cid:90)(cid:90)(cid:90) C × [0 , log Γ( x + z ) dxdydz, I = (cid:90)(cid:90)(cid:90) C × [0 , log Γ( y + z ) dxdydz, (6.4) and I = (cid:90)(cid:90)(cid:90) C × [0 , log Γ( x + y + z ) dxdydz, I = M C = (cid:90)(cid:90) C ( x + y ) dxdy. (6.5) Proof.
Let µ be normalized surface area on ∂B . Then the monomials z a w b , a, b nonneg-ative integers, are orthogonal and (cid:107) z a w b (cid:107) L ( µ ) = a ! b !( a + b + 1)! , see [11, Propositions 1.4.8 and 1.4.9]. Let us estimate Q n = log (cid:89) ( a,b ) ∈ nC a ! b !(1 + a + b )! . We have log (cid:89) ( a,b ) ∈ nC a ! = (cid:88) ( a,b ) ∈ nC log Γ( a + 1) . Recall the multiplication formula for the Gamma function. For Re ( z ) >
0, we haveΓ( nz ) = (2 π ) (1 − n ) / n (2 nz − / Γ( z ) Γ Ç z + 1 n å Γ Ç z + 2 n å · · · Γ Ç z + n − n å . z = ( a + 1) /n , we getlog (cid:89) ( a,b ) ∈ nC a ! = (cid:88) ( a,b ) ∈ nC n (cid:88) k =1 log Γ Ç an + kn å + (cid:88) ( a,b ) ∈ nC − n π + (cid:88) ( a,b ) ∈ nC a −
12 log n. Recalling that d n , the number of elements of nC ∩ N , is the dimension of Poly( nC ),log (cid:89) ( a,b ) ∈ nC a ! = (cid:88) ( a,b ) ∈ nC n (cid:88) k =1 log Γ Ç an + kn å − nd n log 2 π d n π + n log n (cid:88) ( a,b ) ∈ nC Å an ã − d n n. Interpreting the sums over the pairs ( a, b ) as Riemann sums, we get (cid:88) ( a,b ) ∈ nC n (cid:88) k =1 log Γ Ç an + kn å = n I + O ( n ) , (cid:88) ( a,b ) ∈ nC Å an ã = n I + O ( n )with I and I given in (6.4). Together with the estimate d n = n area( C ) + O ( n ), we getlog (cid:89) ( a,b ) ∈ nC a ! = ( n log n ) I + n Ç I − log 2 π C ) å + O ( n log n )where I = (cid:82) C xdxdy . Similarly,log (cid:89) ( a,b ) ∈ nC b ! = ( n log n ) I + n Ç I − log 2 π C ) å + O ( n log n )where I = (cid:82) C ydxdy . Moreover,log (cid:89) ( a,b ) ∈ nC (1 + a + b )! = (cid:88) ( a,b ) ∈ nC n +1 (cid:88) k =2 log Γ Ç a + b + kn å + (cid:88) ( a,b ) ∈ nC − n π + (cid:88) ( a,b ) ∈ nC a + 2 b + 32 log n. = (cid:88) ( a,b ) ∈ nC n +1 (cid:88) k =2 log Γ Ç a + b + kn å + Ç − n å d n log 2 π + n log n (cid:88) ( a,b ) ∈ nC Ç a + bn å + 32 d n log n = ( n log n ) I + n Ç I − log 2 π C ) å + O ( n log n )with I and I given in (6.5). Hence, Q n = n Ç I + I − I − log 2 π C ) å + O ( n log n ) . Now, log δ C ( B ) = lim n →∞ area( C )2 nd n M C Q n - - - - Figure 1: log δ C p ( B ) as a function of p . When p goes to ∞ , log δ C p ( B ) tends to (1 − / (cid:39) − . d n = dim Poly( nC ) (cid:39) n area( C ) and M C = (cid:90)(cid:90) C ( x + y ) dxdy = I . Hence δ C ( B ) = exp Ç I Ç I + I − I − log 2 π C ) åå , which is (6.3). Remark 6.3.
We have log δ C p ( B ) = lim n →∞ area( C p )2 nd n M p Q n,p where d n = dim Poly( nC p ) (cid:39) n area( C p ) and M C p = (cid:90)(cid:90) C p ( x + y ) dxdy = 2 I . Hence δ C p ( B ) = exp Ç p B (1 /p, /p ) Ç I − I − log 2 π p B (1 /p, /p ) åå , where B ( x, y ) denotes the Beta function. Remark 6.4.
The integrals I , I and I can be simplified, eliminating the Gamma func-tion from the integrand. We illustrate this with I . To this end, let F ( x ) := (cid:90) log Γ( x + z ) dz. Then F (cid:48) ( x ) = (cid:90) Γ (cid:48) ( x + z )Γ( x + z ) dz = log Γ( x + 1) − log Γ( x ) = log x. F ( x ) = x (log x − c and it follows from the Raabe integral of the Gamma functionthat c = (cid:82) log Γ( z ) dz = log 2 π . Hence I = (cid:90)(cid:90) C F ( x ) dxdy = (cid:90)(cid:90) C Ç x (log x −
1) + 12 log 2 π å dxdy. In a similar fashion, I = (cid:90)(cid:90) C F ( y ) dxdy = (cid:90)(cid:90) C Ç y (log y −
1) + 12 log 2 π å dxdy and I = (cid:90)(cid:90) C F ( x + y ) dxdy = (cid:90)(cid:90) C Ç ( x + y )[log ( x + y ) −
1] + 12 log 2 π å dxdy. Using these relations and (2.9), we recover (6.2).Making use of (6.2), we get the following result for the case of C = C p , 0 ≤ p < ∞ . Proposition 6.5.
We have log δ C p ( B ) = 3 p B (1 /p, /p ) Ç (cid:90)(cid:90) C p x log xdxdy − (cid:90)(cid:90) C p x log( x + y ) dxdy å , where B ( x, y ) denotes the Beta function. In particular, for p = 1 , p = 2 and p = ∞ weget δ C ( B ) = e − / , δ C ( B ) = √ √ − / √ , δ C ∞ ( B ) = 2 − / e / . Proof.
One hasarea( C p ) = (cid:90)(cid:90) C p dxdy = 12 p B (1 /p, /p ) , I = 13 p B (1 /p, /p ) , and the given formula follows. The particular values for p = 1 , , ∞ follow from computingthe two integrals for these cases. As noted in [9], the results given here in sections 2 and 3 on C − Robin functions and C − transfinite diameter for triangles C in R with vertices (0 , , ( b, , (0 , a ) where a, b are relatively prime positive integers should generalize to the case of a simplex C whichis the convex hull of points { (0 , ..., , ( a , , ..., , ..., (0 , ..., , a d ) } in ( R + ) d with a , ..., a d pairwise relatively prime (using the appropriate definition of the C − Robin function asdefined in Remark 4.5 of [9]). For a product set K = E × · · · × E d in C d where E j arecompact sets in C , Proposition 2.4 of [5] gives that V C,K ( z , ..., z d ) = max[ a g E ( z ) , ..., a d g E d ( z d )]where g E j is the Green function for E j . Hence a generalization of Theorem 4.1 will follow.However, unlike the standard ( C = Σ) case, there is no known nor natural way to expressa formula for the C − extremal function of a product set K when not all of the componentsets are planar compacta; e.g., in the simplest such case, K = E × F ⊂ C with E ⊂ C and F ⊂ C . Nevertheless, it seems that the techniques adopted in sections 5 and 6 usingorthogonal polynomials and/or restricting to d − circled sets could likely be utilized to findmore general product formulas for C − transfinite diameters.17 eferences [1] T. Bayraktar, T. Bloom, N. Levenberg, Pluripotential theory and convex bodies, Mat. Sbornik , (2018), no. 3, 67-101.[2] E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singular-ities, Ann. Inst. Fourier, Grenoble , (1988), no. 4, 133-171.[3] Z. Blocki, A. Edigarian and J. Siciak, On the product property for the transfinitediameter, Ann. Polon. Math. , (2011), no. 3, 209-214.[4] T. Bloom and J.-P. Calvi, On the multivariate transfinite diameter, Annales PoloniciMath. , LXXII.3 (1999), 285-305.[5] L. Bos and N. Levenberg, Bernstein-Walsh theory associated to convex bodies andapplications to multivariate approximation theory,
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