Transfinite diameter notions in C N and integrals of Vandermonde determinants

c 2009 by Institut Mittag-Leffler. All rights reserved

Transfinite diameter notions in C ^N and integrals of Vandermonde determinants

Thomas Bloom and Norman Levenberg

Abstract. We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets inC^N. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter,Math. Ann. 337(2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vander- monde determinants with varying weights for a general class of compact sets and measures inC^N. Our results extend to certain weights and measures defined on cones inR^N.

1. Introduction

Given a compact set E in the complex planeC, thetransfinite diameter of E is the number

d(E) := lim

n→∞ max

ζ1,...,ζn∈E|VDM(ζ1, ..., ζn)|^1/(ⁿ₂) := lim

n→∞ max

ζ1,...,ζn∈E

i<j

|ζi−ζj|^1/(ⁿ₂).

It is well known that this quantity is equal to theChebyshev constant ofE:

T(E) := lim

n→∞

inf

p_nE:p_n(z) =zⁿ+

n−1 j=0

c_jz^j 1/n

(here,p_nE:=sup_z∈E|p(z)|) and also toe^−ρ(E), where ρ(E) := lim

|z|→∞[gE(z)−log|z|]

is the Robin constant of E. The function g_E is the Green function of logarithmic growth associated withE. Moreover, if w is an admissible weight function on E,

Bloom supported in part by an NSERC grant.

weighted versions of the above quantities can be defined. We refer the reader to the book of Saff and Totik [20] for the definitions and relationships.

ForE⊂C^N withN >1, multivariate notions of transfinite diameter, Chebyshev constant and Robin-type constants have been introduced and studied by several people. For an introduction to weighted versions of some of these quantities, see Appendix B by Bloom in [20]. In the first part of this paper (Section2), we discuss a general framework for the various types of transfinite diameters in the spirit of Zaharjuta [22]. In particular, we relate (Theorem 2.9) two weighted transfinite diameters, d^w(E) and δ^w(E), of a compact setE⊂C^N using a remarkable result of Rumely [19] which itself relates the (unweighted) transfinite diameterd(E) with a Robin-like integral formula. Theorem2.9 generalizes a well-known result in one dimension (cf. Theorem 3.1 in Section III.3 of [20]). Recently Robert Berman and S´ebastien Boucksom ([2], [3] and [1]) have given a purely analytic proof of theC^N Rumely formula.

In the second part of the paper (Section3) we generalize to C^N some results on strong asymptotics of Christoffel functions proved in [12] in one variable. For a compact subsetE ofC, an admissible weight functionwonE, and a positive Borel measureμonEsuch that the triple (E, w, μ) satisfies a weighted Bernstein–Markov inequality (see (3.5)), we take, for eachn=1,2, ..., a set of orthonormal polynomials q⁽ⁿ⁾₁ , ..., qn⁽ⁿ⁾ with respect to the varying measures w(z)²ⁿdμ(z), where degq_j⁽ⁿ⁾= j−1, and form the sequence of Christoffel functionsKn(z):=n

j=1|q⁽ⁿ⁾_j (z)|². In [12]

we showed that

(1.1) 1

nK_n(z)w(z)²ⁿdμ(z)→dμ^w_eq(z)

weak-*, whereμ^w_eqis the potential-theoretic weighted equilibrium measure. The key ingredients to proving (1.1) are, firstly, the verification that

(1.2) lim

n→∞Z_n^1/n2=δ^w(E), where

(1.3)

Zn=Zn(E, w, μ) :=

Eⁿ

|VDM(λ1, ..., λn)|²w(λ1)²ⁿ...w(λn)²ⁿdμ(λ1)...dμ(λn);

and, secondly, a “large deviation” result in the spirit of Johansson [16]. We general- ize these two results toC^N,N >1 (Theorems3.1and3.2). The methods are similar to the corresponding one-variable methods and were announced in [12], Remark 3.1.

In particular,δ^w(E) is interpreted as the transfinite diameter of a circled set in one

dimension higher. We also discuss the case whereE=Γ is an unbounded cone in R^N for special weights and measures.

Berman, Boucksom and Nystrom ([1]–[6]) have studied wide-ranging general- izations of notions related to strong asymptotics for compact subsets of C^N. We end the paper with a short section which describes some of their results and the relation to the topics of this paper. We are grateful to the authors for making their references available to us. The second author would also like to thank Sione Ma’u and Laura DeMarco for helpful conversations.

2. Transfinite diameter notions inC^N

We begin by considering a function Y from the set of multiindices α∈N^N to the nonnegative real numbers satisfying

(2.1) Y(α+β)≤Y(α)Y(β) for allα, β∈N^N.

We call a function Y satisfying (2.1) submultiplicative; we have three main ex- amples below. Let e1(z), ..., ej(z), ... be a listing of the monomials {ei(z)=z^α(i)= z^α₁¹...z_N^αN} in C^N indexed using a lexicographic ordering on the multiindices α=

α(i)=(α₁, ..., α_N)∈N^N, but with dege_i=|α(i)| nondecreasing. We write |α|:=

N j=1α_j.

We define the following integers:

(1) m^(N)_d =md:= the number of monomials ei(z) of degree at most d in N variables;

(2) h^(N_d ⁾=hd:= the number of monomialsei(z) of degree exactly din N vari- ables;

(3) l_d^(N)=ld:= the sum of the degrees of themd monomialsei(z) of degree at mostdin N variables.

We have the following relations:

(2.2) m^(N)_d = N+d

d

and h^(N_d ⁾=m^(N)_d −m^(N)_d−1=

N−1+d d

, and

(2.3) h^(N+1)_d = N+d

d

=m^(N)_d and l^(N_d ⁾=N N+d

N+1

= N

N+1dm^(N)_d . The elementary fact that the dimension of the space of homogeneous polynomials of degree din N+1 variables equals the dimension of the space of polynomials of degree at mostdin N variables will be crucial. Finally, we let

r_d^(N)=rd:=dh^(N_d ⁾=d(m^(N_d ⁾−m^(N_d−⁾₁)

which is the sum of the degrees of thehd monomialsei(z) of degree exactlydinN variables. We observe that

(2.4) l^(N_d ⁾=

d k=1

r^(N)_k = N k=1

kh^(N_k ⁾.

LetK⊂C^N be compact. Here are our three natural constructions associated withK.

(1) Chebyshev constants: Define the class of polynomials Pi=P(α(i)) :=

ei(z)+

j<i

cjej(z)

; and the Chebyshev constant

Y1(α) := inf{pK:p∈Pi}.

We write tα,K:=tα(i),K for a Chebyshev polynomial; i.e., tα,K∈P(α(i)) and tα,KK=Y1(α).

(2) Homogeneous Chebyshev constants: Define the class of homogeneous poly- nomials

P_i^(H)=P^(H)(α(i)) :=

e_i(z)+

j<i deg(ej)=deg(ei)

c_je_j(z)

;

and the homogeneous Chebyshev constant

Y2(α) := inf{pK:p∈P_i^(H)}.

We writet^(H)_α,K:=t^(H)_α(i),K for a homogeneous Chebyshev polynomial; i.e., t^(H)_α,K∈P^(H)(α(i)) and t^(H)_α,KK=Y2(α).

(3) Weighted Chebyshev constants: Letwbe an admissible weight function on K (see below) and let

Y3(α) := inf{w^|α(i)|pK:p∈Pi}

be the weighted Chebyshev constant. Note that we use the polynomial classesP_i as in (1). We write t^w_α,K for a weighted Chebyshev polynomial; i.e.,t^w_α,K is of the formw^α(i)pwithp∈P(α(i)) andt^w_α,KK=Y3(α).

Let Σ denote the standard (N−1)-simplex in R^N; i.e., Σ =

θ= (θ₁, ..., θ_N)∈R^N: N j=1

θ_j= 1 andθ_j≥0, j= 1, ..., N

,

and let

Σ⁰:={θ∈Σ :θj>0, j= 1, ..., N}.

Given a submultiplicative functionY(α), define, as with the above examples, a new function

(2.5) τ(α) :=Y(α)^1/|α|.

An examination of Lemmas 1, 2, 3, 5 and 6 in [22] shows that (2.1) is the only property of the numbersY(α) needed to establish those lemmas. That is, we have the following results for Y:N^N→R⁺ satisfying (2.1) and the associated function τ(α) in (2.5).

Lemma 2.1. For allθ∈Σ⁰, the limit T(Y, θ) := lim

α/|α|→θ

Y(α)^1/|α|= lim

α/|α|→θ

τ(α)

exists.

Lemma 2.2. The function θ→T(Y, θ)is log-convex on Σ⁰(and hence contin- uous).

Lemma 2.3. Given b∈∂Σ, lim inf

θ→b θ∈Σ⁰

T(Y, θ) = lim inf

i→∞

α(i)/|α(i)|→b

τ(α(i)).

Lemma 2.4. Let θ(k):=α(k)/|α(k)| for k=1,2, ... and let Q be a compact subset of Σ⁰. Then

lim sup

|α|→∞{logτ(α(k))−logT(Y(θ(k))) :|α(k)|=αandθ(k)∈Q}= 0.

Lemma 2.5. Define τ(Y) := exp

1 meas(Σ) _Σ0

logT(Y, θ)dθ

.

Then

lim

d→∞

1 hd

|α|=d

logτ(α) = logτ(Y);

i.e.,using (2.5),

lim

d→∞

|α|=d

Y(α) 1/dh_d

=τ(Y).

One can incorporate all of theY(α)’s for|α|≤d; this is the content of the next result.

Theorem 2.6. We have lim

d→∞

|α|≤d

Y(α) 1/l_d

exists and equalsτ(Y).

Proof. Define the geometric means τ_d⁰:=

|α|=d

τ(α) 1/hd

, d= 1,2, ... . The sequence

logτ₁⁰,logτ₁⁰, ...(r1 times), ...,logτ_d⁰,logτ_d⁰, ...(rd times), ...

converges to logτ(Y) by the previous lemma; hence the arithmetic mean of the first ld=d

k=1rk terms (see (2.4)) converges to logτ(Y) as well. Exponentiating this arithmetic mean gives

(2.6)

d k=1

(τ_k⁰)^rk 1/ld

= d

k=1

|α|=k

τ(α)^k 1/ld

=

|α|≤d

Y(α) 1/ld

and the result follows.

Returning to our examples (1)–(3), example (1) was the original setting of Zaharjuta [22] which he utilized to prove the existence of the limit in the definition of thetransfinite diameter of a compact setK⊂C^N. Forζ₁, ..., ζ_n∈C^N, let

VDM(ζ1, ..., ζn) = det[ei(ζj)]i,j=1,...,n

(2.7)

= det

⎛

⎜⎝

e₁(ζ₁) e₁(ζ₂) ... e₁(ζ_n) ... ... ... ...

e_n(ζ₁) e_n(ζ₂) ... e_n(ζ_n)

⎞

⎟⎠

and for a compact subsetK⊂C^N let V_n=V_n(K) := max

ζ₁,...,ζ_n∈K|VDM(ζ₁, ..., ζ_n)|. Then

(2.8) d(K) = lim

d→∞V_m^1/l_d^d

is the transfinite diameter of K; Zaharjuta [22] showed that the limit exists by showing that one has

(2.9) d(K) = exp

1 meas(Σ) _Σ0

logτ(K, θ)dθ

,

where τ(K, θ)=T(Y1, θ) from (1); i.e., the right-hand-side of (2.9) is τ(Y1). This follows from Theorem2.6forY=Y1 and the estimate

d k=1

(τ_k⁰)^rk 1/ld

≤V_m^1/l_d^d≤(md!)^1/ld d

k=1

(τ_k⁰)^rk 1/ld

in [22] (compare (2.6)).

For a compactcircled setK⊂C^N; i.e.,z∈K if and only ife^iφz∈K,φ∈[0,2π], one need only consider homogeneous polynomials in the definition of the directional Chebyshev constantsτ(K, θ). In other words, in the notation of (1) and (2),Y₁(α)=

Y2(α) for allαso that

T(Y₁, θ) =T(Y₂, θ) for circled sets K.

This is because for such a set, if we write a polynomialpof degreedasp=d j=0H_j, whereH_j is a homogeneous polynomial of degreej, then, from the Cauchy integral formula, HjK≤pK, j=0, ..., d. Moreover, a slight modification of Zaharjuta’s arguments prove the existence of the limit of appropriate roots of maximal ho- mogeneous Vandermonde determinants; i.e., the homogeneous transfinite diameter d^(H)(K) of a compact set (cf. [15]). From the above remarks, it follows that (2.10) d(K) =d^(H)(K) for circled sets K.

Since we will be using the homogeneous transfinite diameter, we amplify the discus- sion. We relabel the standard basis monomials{e^(H,d)_i (z)=z^α(i)=z₁^α1...z^α_N^N}, where

|α(i)|=d,i=1, ..., hd, we define thed-homogeneous Vandermonde determinant (2.11) VDMHd(ζ1, ..., ζh_d) := det

e^(H,d)_i (ζj)

i,j=1,...,h_d.

Then

(2.12) d^(H)(K) = lim

d→∞

max

ζ₁,...,ζ_hd∈K|VDMH_d(ζ₁, ..., ζ_hd)|1/dh_d

is thehomogeneous transfinite diameter of K; the limit exists and equals exp

1 meas(Σ) _Σ0

logT(Y₂, θ)dθ

,

whereT(Y2, θ) comes from (2).

Finally, related to example (3), there are similar properties for the weighted ver- sion of directional Chebyshev constants and transfinite diameter. To define weighted notions, let K⊂C^N be closed and let w be an admissible weight function on K;

i.e., wis a nonnegative, upper semicontinuous function with {z∈K:w(z)>0} non- pluripolar. Let Q:=−logw and define the weighted pluricomplex Green function V_K,Q^∗ (z):=lim sup_ζ→zV_K,Q(ζ), where

VK,Q(z) := sup{u(z) :u∈L(C^N) andu≤QonK}.

Here,L(C^N) is the set of all plurisubharmonic functionsuonC^N with the property thatu(z)−log|z|=O(1), as|z|→∞. IfKis closed but not necessarily bounded, we require thatwsatisfies the growth property

(2.13) |z|w(z)→0, as |z|→∞, z∈K,

so thatVK,Q is well-defined and equalsVK∩B_R,Q forR>0 sufficiently large, where BR={z:|z|≤R} (Definition 2.1 and Lemma 2.2 of Appendix B in [20]). The un- weighted case is whenw≡1 (Q≡0); we then writeV_K for the pluricomplex Green function. A compact setK is calledregular ifVK=V_K^∗; i.e.,VK is continuous; and Kislocally regular if for eachz∈K, the setsK∩B(z, r) are regular forr>0, where B(z, r) denotes the ball of radiusrcentered atz. We define theweighted transfinite diameter

d^w(K) := exp 1

meas(Σ) _Σ0

logτ^w(K, θ)dθ

as in [11], whereτ^w(K, θ)=T(Y3, θ) from (3); i.e., the right-hand-side of this equa- tion is the quantityτ(Y3).

We remark for future use that if {Kj}^∞j=1 is a decreasing sequence of locally regular compact sets withKj↓K, and ifwj is a continuous admissible weight func- tion onK_j withw_j↓wonK, wherewis an admissible weight function onK, then the argument in Proposition 7.5 of [11] shows that lim_j→∞τ^wj(Kj, θ)=τ^w(K, θ) for

allθ∈Σ⁰ (we mention that there is a misprint in the statement of this proposition in [11]) and hence

(2.14) lim

j→∞d^wj(Kj) =d^w(K).

In particular, (2.14) holds in the unweighted case (w≡1) for any decreasing sequence {Kj}^∞j=1 of compact sets withKj↓K; i.e.,

(2.15) lim

j→∞d(Kj) =d(K) (cf. [11], equation (1.13)). Another useful fact is that

(2.16) d(K) =d(K) and d^(H)(K) =d^(H)(K) for compactK, where

K:={z∈C^N:|p(z)| ≤ pK for all polynomialsp} is thepolynomial hull ofK.

Another natural definition of a weighted transfinite diameter uses weighted Vandermonde determinants. Let K⊂C^N be compact and let w be an admissible weight function onK. Given ζ₁, ..., ζ_n∈K, let

W(ζ1, ..., ζn) := VDM(ζ1, ..., ζn)w(ζ1)^|α(n)|...w(ζn)^|α(n)|

= det

⎛

⎜⎝

e1(ζ1) e1(ζ2) ... e1(ζn) ... ... ... ...

en(ζ1) en(ζ2) ... en(ζn)

⎞

⎟⎠w(ζ1)^|α(n)|...w(ζn)^|α(n)|

be aweighted Vandermonde determinant. Let

(2.17) Wn:= max

ζ₁,...,ζ_n∈K|W(ζ1, ..., ζn)|

and define ann-th weighted Fekete setforKandwto be a set ofnpointsζ₁, ..., ζ_n∈ K with the property that

|W(ζ1, ..., ζn)|= sup

ξ1,...,ξn∈K

|W(ξ1, ..., ξn)|. Also, define

(2.18) δ^w(K) := lim sup

d→∞ W_m^1/ld

d .

We will show in Proposition 2.7 that lim_d→∞Wm^1/l_d^d (the weighted analogue of (2.8)) exists. The question of the existence of this limit if N >1 was raised

in [11]. Moreover, using a recent result of Rumely, we will show in Theorem 2.9 howδ^w(K) is related tod^w(K):

(2.19) δ^w(K) =

exp

−

K

Q

dd^cV_K,Q^∗ N1/N

d^w(K),

where (dd^cV_K,Q^∗ )^N is the complex Monge–Amp`ere operator applied to V_K,Q^∗ . We refer the reader to [17] or Appendix B of [20] for more on the complex Monge–

Amp`ere operator.

We begin by proving the existence of the limit in the definition of δ^w(E) in (2.18) for a setE⊂C^N and an admissible weightw onE (see also [2]).

Proposition 2.7. LetE⊂C^N be a compact set with an admissible weight func- tionw. The limit

δ^w(E) := lim

d→∞

max

λ⁽ⁱ⁾∈E|VDM(λ⁽¹⁾, ..., λ^{(m(N)d )})|w(λ⁽¹⁾)^d...w(λ^{(m(N)d )})^d 1/l^(N)_d

exists.

Proof. Following [8], we define the circled set

F=F(E, w) :={(t, z) = (t, tλ)∈C^N+1:λ∈E and|t|=w(λ)}.

We first relate weighted Vandermonde determinants forE with homogeneous Van- dermonde determinants for the compact set

(2.20) F(D) :={(t, z) = (t, tλ)∈C^N+1:λ∈E and|t| ≤w(λ)}.

Note that F⊂F⊂F(D)⊂F (cf. [8], (2.4)) where F is the polynomial hull of F (recall (2.16)); thus

(2.21) d^(H)(F) =d^(H)(F(D)).

To this end, for each positive integerd, choose m^(N_d ⁾=

N+d d

(recall (2.2)) points{(ti, z⁽ⁱ⁾)}_i=1,...,m^(N)

d

={(ti, tiλ⁽ⁱ⁾)}_i=1,...,m^(N)

d

inF(D) and form thed-homogeneous Vandermonde determinant

VDMHd((t1, z⁽¹⁾), ...,(t_m(N) d

, z^{(m(N)d )})).

We extend the lexicographical order of the monomials in C^N to C^N+1 by letting t precede any of z1, ..., zN. Writing the standard basis monomials of degree d in C^N+1 as

{t^d−je^(H,d)_k (z) :j= 0, ..., dandk= 1, ..., h_j};

i.e., for each powerd−joftwe multiply by the standard basis monomials of degree j in C^N, and dropping the superscript (N) in m^(N_d ⁾, we have the d-homogeneous Vandermonde matrix

⎛

⎜⎜

⎜⎜

⎝

t^d₁ t^d₂ ... t^d_m

d

t^d₁⁻¹e2(z⁽¹⁾) t^d₂⁻¹e2(z⁽²⁾) ... t^d_m⁻_d¹e2(z^(md))

... ... ... ...

em_d(z⁽¹⁾) em_d(z⁽²⁾) ... em_d(z^(md))

⎞

⎟⎟

⎟⎟

⎠

=

⎛

⎜⎜

⎜⎜

⎝

t^d₁ t^d₂ ... t^d_m

d

t^d₁⁻¹z⁽¹⁾₁ t^d₂⁻¹z₁⁽²⁾ ... t^d_m⁻¹

dz₁^(md)

... ... ... ...

(z_N⁽¹⁾)^d (z⁽²⁾_N )^d ... (z_N^(md))^d

⎞

⎟⎟

⎟⎟

⎠.

Factoringt^d_i out of theith column, we obtain

VDMHd((t1, z⁽¹⁾), ...,(tm_d, z^(md))) =t^d₁...t^d_mdVDM(λ⁽¹⁾, ..., λ^(md));

thus, writing|A|:=|detA|for a square matrixA,

t^d₁ t^d₂ ... t^d_m

d

t^d₁⁻¹z⁽¹⁾₁ t^d₂⁻¹z⁽²⁾₁ ... t^d_m⁻¹

dz₁^(md)

... ... ... ...

(z_N⁽¹⁾)^d (z⁽²⁾_N )^d ... (z^(m_N ^d))^d (2.22)

=|t1|^d...|tm_d|^d

1 1 ... 1

λ⁽¹⁾₁ λ⁽²⁾₁ ... λ^(m₁ ^d) ... ... ... ...

(λ⁽¹⁾_N )^d (λ⁽²⁾_N )^d ... (λ^(m_N ^d))^d ,

whereλ^(j)_k =z_k^(j)/t_j providedt_j=0. By definition ofF(D), as (t_i, z⁽ⁱ⁾)=(t_i, t_iλ⁽ⁱ⁾)∈ F(D), we have|t_i|≤w(λ⁽ⁱ⁾). Clearly the maximum of

|VDMHd((t1, z⁽¹⁾), ...,(tmd, z^(md)))|

over points inF(D) will occur when all|tj|=w(λ^(j))>0 (recall thatwis an admis- sible weight) so that from (2.22),

max

(t_i,z⁽ⁱ⁾)∈F(D)|VDMH_d((t₁, z⁽¹⁾), ...,(t_md, z^(md)))|

= max

λ⁽ⁱ⁾∈E|VDM(λ⁽¹⁾, ..., λ^(md))|w(λ⁽¹⁾)^d...w(λ^(md))^d. As mentioned in the discussion of (2.12) the limit

lim

d→∞

max

(t_i,z⁽ⁱ⁾)∈F(D)|VDMHd((t1, z⁽¹⁾), ...,(tm_d, z^(md)))|1/dh^(N+1)_d

=:d^(H)(F(D)) exists [15]; thus the limit

dlim→∞

max

λ⁽ⁱ⁾∈E|VDM(λ⁽¹⁾, ..., λ^(md))|w(λ⁽¹⁾)^d...w(λ^(md))^d 1/l^(N)_d

:=δ^w(E) exists.

Corollary 2.8. For a nonpluripolar compact set E⊂C^N with an admissible weight functionw and

(2.23)

F=F(E, w):={(t, z)=(t, tλ)∈C^N+1:λ∈E and|t|=w(λ)}, δ^w(E)=d^(H)(F)^(N+1)/N=d(F)^(N+1)/N.

Proof. The first equality follows from the proof of Proposition2.7using (2.21) and the relation

l_d^(N)= N

N+1dh^(N_d ⁺¹⁾ (see (2.3)). The second equality is (2.10).

We next relate δ^w(E) and d^w(E) but we first recall the remarkable formula of Rumely [19] (see also [2]). For a plurisubharmonic function uin L(C^N) we can define theRobin function associated with u:

ρu(z) := lim sup

|λ|→∞[u(λz)−log|λ|].

This function is plurisubharmonic (cf. [7], Proposition 2.1) and logarithmically ho- mogeneous:

ρu(tz) =ρu(z)+log|t| fort∈C.

Foru=V_E,Q^∗ (V_E^∗) we writeρu=ρE,Q (ρE). Rumely’s formula relatesρE andd(E):

−logd(E) = 1

N _CN−1ρE(1, t2, ..., tN)(dd^cρE(1, t2, ..., tN))^N−1 (2.24)

+

C^N−2

ρ_E(0,1, t₃, ..., t_N)(dd^cρ_E(0,1, t₃, ..., t_N))^N−2+...

+

C

ρ_E(0, ...,0,1, t_N)(dd^cρ_E(0, ...,0,1, t_N))+ρ_E(0, ...,0,1)

.

Here we make the convention thatdd^c=(i/π)∂∂so that in any dimensiond=1,2, ...,

C^d

(dd^cu)^d= 1

for anyu∈L⁺(C^d); i.e., for any plurisubharmonic functionuin C^d which satisfies C₁+log(1+|z|)≤u(z)≤C₂+log(1+|z|)

for someC1 andC2.

We begin by rewriting (2.24) forregular circled setsE using an observation of Sione Ma’u. Note that for such sets,V_E^∗=ρ⁺_E:=max(ρE,0) (cf. [13], Lemma 5.1). If we intersectEwith a hyperplaneHthrough the origin, e.g., by rotating coordinates, we take H={z=(z1, ..., zN)∈C^N:z1=0}, then E∩H is a regular, compact, circled set inC^N−1 (which we identify withH). Moreover, we have

ρ_H∩E(z₂, ..., z_N) =ρ_E(0, z₂, ..., z_N)

since each side is logarithmically homogeneous and vanishes for (z2, ..., zN)∈

∂(H∩E). Thus the terms

C^N−2ρE(0,1, t3, ..., tN)(dd^cρE(0,1, t3, ..., tN))^N−2 +...+

C

ρE(0, ...,0,1, tN)(dd^cρE(0, ...,0,1, tN))+ρE(0, ...,0,1) in (2.24) are seen to equal

(N−1)d^CN−1(H∩E)

(where we temporarily writed^CN−1 to denote the transfinite diameter inC^N−1 for emphasis) by applying (2.24) inC^N−1 to the setH∩E. Hence we have

−logd(E) = 1 N _CN−1

ρE(1, t2, ..., tN)(dd^cρE(1, t2, ..., tN))^N−1 (2.25)

+N−1

N [−logd^CN−1(H∩E)].

Theorem 2.9. For a nonpluripolar compact set E⊂C^N with an admissible weight functionw,

(2.26) δ^w(E) =

exp

−

E

Q(dd^cV_E,Q^∗ )^N 1/N

d^w(E).

Proof. We first assume that E is locally regular and Q is continuous. It is known in this case that VE,Q=V_E,Q^∗ (cf. [21], Proposition 2.16). As before, we define the circled set

F=F(E, w) :={(t, z) = (t, tλ)∈C^N+1:λ∈E and|t|=w(λ)}.

We claim that this is a regular compact set; i.e., that VF is continuous. First of all,V_F^∗(t, z)=max(ρF(t, z),0) (cf. Proposition 2.2 of [8]) so that it suffices to verify thatρ_F(t, z) is continuous. From Theorem 2.1 and Corollary 2.1 of [8],

(2.27) VE,Q(λ) =ρF(1, λ) onC^N

which implies, by the logarithmic homogeneity ofρF, thatρF(t, z) is continuous on C^N+1\{(t, z):t=0}. Corollary 2.1 and (2.6) in [8] give that

(2.28) ρ_F(0, λ) =ρ_E,Q(λ) forλ∈C^N

and ρE,Q is continuous by Theorem 2.5 of [11]. Moreover, the limit exists in the definition of ρE,Q:

ρE,Q(λ) := lim sup

|t|→∞ [VE,Q(tλ)−log|t|] = lim

|t|→∞[VE,Q(tλ)−log|t|];

and the limit is uniform inλ (cf. Corollary 4.4 of [13]) which implies, from (2.27) and (2.28), that limt→0ρF(t, λ)=ρF(0, λ) so that ρF(t, z) is continuous. In partic- ular, from Theorem 2.5 in Appendix B of [20],

VE,Q(λ) =Q(λ) =ρF(1, λ) on the support of (dd^cVE,Q)^N so that

(2.29)

E

Q(λ)(dd^cV_E,Q(λ))^N=

C^N

ρ_F(1, λ)(dd^cρ_F(1, λ))^N.

On the other hand, E_ρ^w:={λ∈C^N:ρE,Q(λ)≤0} is a circled set, and, according to (3.14) in [11],d^w(E)=d(E_ρ^w). But

ρE,Q(λ) = lim sup

|t|→∞ [VE,Q(tλ)−log|t|]

= lim sup

|t|→∞ [ρF(1, tλ)−log|t|] = lim sup

|t|→∞ ρF(1/t, λ) =ρF(0, λ).

Thus

E_ρ^w={λ∈C^N:ρ_F(0, λ)≤0}=F∩H, whereH={(t, z)∈C^N+1:t=0}and hence

(2.30) d^w(E) =d(E_ρ^w) =d(F∩H).

From (2.25) applied to F⊂C^N+1, (2.31) −logd(F) = 1

N+1 _CN

ρF(1, λ)(dd^cρF(1, λ))^N+ N

N+1[−logd(F∩H)].

Finally, from (2.23) (note thatF=F sincewis continuous),

(2.32) δ^w(E) =d(F)^(N+1)/N;

putting together (2.29), (2.30), (2.31) and (2.32) gives the result if E is locally regular andQis continuous.

The general case follows from approximation. Take a sequence of locally regular compact sets{Ej}^∞j=1 decreasing toE and a sequence of weight functions{wj}^∞j=1

withwj continuous and admissible onEj and wj↓w on E (cf. Lemma 2.3 of [8]).

From (2.14) we have

(2.33) lim

j→∞d^wj(E_j) =d^w(E).

Also, by Corollary2.8we have

(2.34) δ^wj(Ej) =d(Fj)^(N+1)/N, where

Fj=Fj(Ej, wj) ={t(1, λ) :λ∈Ej and|t|=wj(λ)}. SinceEj+1⊂Ej andwj+1≤wj, the sets

Fj(D) ={t(1, λ) :λ∈Ej and|t| ≤wj(λ)} satisfyFj+1(D)⊂Fj(D) and hence

d(Fj+1(D)) =d(Fj+1)≤d(Fj(D)) =d(Fj).

SinceFj(D)↓F(D) (see (2.20)), we conclude from (2.15), (2.21), (2.10) and (2.34) that

(2.35) lim

j→∞δ^wj(E_j) =δ^w(E).

Applying (2.26) toEj, wj, Qj and using (2.33) and (2.35), we conclude that

E_j

Qj

dd^cVEj,Qj

N

→

E

Q

dd^cV_E,Q^∗ N

,

from Lemma 3.6.2 of [17] completing the proof of (2.26).

3. Integrals of Vandermonde determinants

In this section, we first state and prove the analogue of an “unweighted” gen- eralization to C^N of Theorem 2.1 of [12] as it has a self-contained proof utilizing Chebyshev polynomials discussed in Section 2. We first recall some terminology.

Given a compact set E⊂C^N and a measure ν on E, we say that (E, ν) satisfies a Bernstein–Markov inequality for holomorphic polynomials in C^N if, given ε>0, there exists a constant M=M(ε) such that for all such polynomials Q_n of degree at mostn,

(3.1) QnE≤M(1+ε)ⁿQnL²(ν).

Theorem 3.1. Let (E, μ)satisfy a Bernstein–Markov inequality. Then lim

d→∞Z_d(E, μ)^1/2l(N)d =d(E), where

(3.2) Zd(E, μ) :=

E^m

(N) d

|VDM(λ⁽¹⁾, ..., λ^m(N)d )|²dμ(λ⁽¹⁾)...dμ(λ^m(N)d ).

Proof. Since VDM(ζ1, ..., ζn)=det[ei(ζj)]i,j=1,...,n for any positive integern, if we apply the Gram–Schmidt procedure to the monomials e1, ..., e_m(N)

d

to obtain orthogonal polynomials q1, ..., q_m(N)

d

with respect to μ, where qj∈Pj has minimal L²(μ)-norm among all such polynomials, we get, upon using elementary row opera- tions on VDM(ζ₁, ..., ζ

m^(N)_d ) and expanding the determinant (cf. [14], Chapter 5 or Section 2 of [12])

(3.3)

E^m

(N) d

|VDM(ζ1, ..., ζ_m(N) d

)|²dμ(ζ1)...dμ(ζ_m(N) d

) =m^(N_d ⁾!

m^(N)_d

j=1

qj²L²(μ).

Let tα,E∈P(α) be a Chebyshev polynomial; i.e., tα,EE=Y1(α). Then Theo- rem2.6shows that

lim

d→∞

|α|≤d

t_α,EE

1/l_d

=τ(Y₁)

since

lim

d→∞(m^(N)_d !)^1/l(N)d = 1.

Zaharjuta’s theorem (2.9) shows thatτ(Y₁)=d(E) so we need to show that

(3.4) lim

d→∞

|α|≤d

tα,EE

1/l_d

= lim

d→∞

|α|≤d

qαL²(μ)

1/l_d

.

This follows from the Bernstein–Markov property. First note that qαL²(μ)≤ tα,EL²(μ)≤μ(E)tα,EE

from the L²(μ)-norm minimality of qα; then, given ε>0, the Bernstein–Markov property and the sup-norm minimality oft_α,E give

tα,EE≤ qαE≤M(1+ε)^|α|qαL²(μ)

for some M=M(ε)>0. Taking products of these inequalities over |α|≤d; taking l_d-th roots; and letting ε→0 gives the result. This reasoning is adapted from the proof of Theorem 3.3 in [9].

A weighted polynomial onEis a function of the formw(z)ⁿpn(z), wherepnis a holomorphic polynomial of degree at mostn. Letμbe a measure with support inE such that (E, w, μ) satisfies a Bernstein–Markov inequality for weighted polynomials (referred to as aweighted B–M inequality in [8]): givenε>0, there exists a constant M=M(ε) such that for all weighted polynomialswⁿpn,

(3.5) wⁿp_nE≤M(1+ε)ⁿwⁿp_nL²(μ). Generalizing Theorem3.1, we have the following result.

Theorem 3.2. Let (E, w, μ)satisfy a Bernstein–Markov inequality (3.5) for weighted polynomials. Then

lim

d→∞Z^1/2l

(N) d

d =δ^w(E), where

Z_d=Z_d(E, w, μ) :=

E^m

(N) d

|VDM(λ⁽¹⁾, ..., λ^{(m(N)d )})|² (3.6)

×w(λ⁽¹⁾)^2d...w(λ^{(m(N)d )})^2ddμ(λ⁽¹⁾)...dμ(λ^{(m(N)d )}).