Ark. Mat., 39 (2001), 245-262
@ 2001 by Institut Mittag-Leffler.
Ali rights reserved
On the Siciak extremal function
for real compact convex sets
Len Bos, Jean-Paul Calvi and Norman Levenberg
1. Introduction
Let E be a bounded Borel set in eN. Define
(1.1)
VE(Z) :=sup{u(z) :UEL, u:::;Oon E},
where
L:={u plurisubharmonic in eN :u(z) :::;log+Izl+C for some C}
is the class of plurisubharmonic
functions of logarithmic growth (here we have
Izl=(L~=1IzjI2)1/2 and log+ Izl=max{O,loglzl}).
Then the upper semicontinuous regularization VÊ(z):=limsuPC-+z VE(() is called the (Siciak) extremal function
of E. If K is a compact set in eN, then the extremal function in (1.1) can be gotten
via the formula
(1.2)
VK(Z) :=max{ o,suP{ de~p log Ip(z)l:p holomorphic polynomial, IIpIIK:::;1} }
(Theorem 5.1.7 in [KI]). Here, IlpIlK:=SUPzEKIp(z)1 denotes the uniform norm
on K. We say that K is regular if and only if V;=VK. Note that if we let
R:= {z E eN : Ip(Zl, ... , ZN) 1:::;IIpllKfor all polynomials p}
denote the polynomial hull of K, then
1. R ={ZEeN :VK(z)=O};
2. VR=VK,
246
Len Bos, Jean-Paul
Calvi and Norman Levenberg
For future use, we say that K is polynomially convex if K =-K.
ln one complex variable (N = 1), the function VK (or, in general, V;), is the
classical Green function of the planar compact set K with logarithmic pole at infinity. The theory of conformaI mapping can be used to find explicit formulas for
VK in many cases. We recall the case of an interval: since h( 0 :=( + yi (2 -1 is
a conformaI map of the complement of the interval [-1,1] onto the complement
of the closed unit disk, we have V[-l,l](()=log Ih(OI. ln eN for N>l, examples
of explicit (or even semi-explicit!) formulas for VK are severely lacking. The first
interesting formulas, due to Siciak, dealt with product sets and circled sets (cf., [S]).
ln [L], [BI] and [B2], Lundin and Baran have given simplifications of formula (1.2)
in the case where K is a convex set in RN -considered as a subset of eN -which
is symmetric with respect to the origin, i.e., xEK implies -xEK.
EN is the closed unit ball in RN, Le.,
For example, if
EN:= {z E eN: Imz1 =... = lm ZN = 0, (Re Zl)2+...+(Re ZN)2 :S 1},
then VEN(z) = ~ log h(lzl2 + Iz2-11), where z2 =zf + ...+z;. (note El = [-1,1]). A
few more explicit examples can be obtained using the following result of Klimek.
Proposition
1.1. ([KI]) Let f=(fI, ... ,fN) be a polynomial mapping of eN
into eN with the properties that deg fI =...=deg fN :=d;:::1 and ]-1 (0) = {O}(where
]:= (JI, ... , ] N) denotes the top degree (d) homogeneous piece of f). Then for any
compact set K,
1
Vf-l(K)(Z)
= dVK(f(z)).
ln e2, if we set f(Zl, Z2):=(zf, z?), and if we take K =52, where
52:= {(X1,X2) E R2 :X1, X2;:::0, Xl +X2:S 1}
is the standard
simplex, then f-1(52)=E2
VS2 (Zl, Z2)
= log
and we obtain
h(lz11+lz21+
IZ1+Z2 -11).
We will use this fact later in the paper.
ln the examples of EN and 52, the extremal functions were gotten from onevariable functions. More generally, the following statement is a consequence of the
results of Baran and Lundin.
Proposition 1.2. ([BI], [B2], [B3], [L]) Let KCRN be a convex body (i.e.,
a compact, convex set with non-empty interior in RN) which is symmetric with
respect to the origin. Then for aU z EeN ,
(1.3)
VK(z) = 1I(z) :=sup{V{(K)(l(z)): l E RN*}.
"
On the Siciak extremal function for real compact convex sets
247
Here RN* is the set of aIl non-zero linear functionals l on RN, i.e., l ER N* is a
real-linear mapping from RN to R. We can consider each lERN* as an element in
CN* the (complex) vector space of complex-linear functionals on CN via l(x+iy)=
l(x)+il(y).
Our original goal was to determine whether (1.3) is valid if the symmetry
hypothesis is omitted. Using the example of the standard simplex 32 C C2, it is
not tao hard to see that the answer is no. However, equality in (1.3) does remain
valid for real convex compact sets, symmetric or not, at every real point, i.e., for
each z=xERN cCN. Indeed, more is true, see Corollary 3.2. The key idea is a
geometric property of convex sets due to Kroo and Schmidt [KR]; we study this
property in detail in the next section. This suggests a more general question: Let
N> 1 and suppose K C CN is compact. Let
(1.4)
V(Z):=sup{VZ(K)(l(z)):lECN*,
l;iD},
i.e., l is permitted to vary over all non-zero complex-linear functionals on CN.
When do we get equality in (1.3) if V is replaced by V? ln Section 4, we discuss
more general situations when the computation of VK can be reduced to one-variable
calculations; in particular, we show that if K is polynomially convex (K =K) and
V (z) = VK (z) in CN, then K must be lineally convex, i.e. the complement of K
is the union of complex hyperplanes. ln Section 5 we show that for the simplex
32cC2, V*;iVs2 (here, V*(z)=limsup(-tz V(()). This involves a detailed study of
the Robin functions associated to V and Vs2; these objects play a vital role in the
study of functions in the class L if N> 1. We conclude the paper in Section 6 by
showing that among the regular, polynomially convex and lineally convex compact
sets K in C2, the ones for which V*;i VK form a "large" class.
Remark. Note that if we replace l by a scalar multiple tl, then Vtl(K)otl=
VZ(K)ol. Thus considering upper envelopes over aIl linear functionals or simply,
e.g., over aIllinear functionals normalized to have norm 1, yield the same functions
V and V. Similarly, if lECN* and aEC is constant, then vel+a)(K)((l+a)(z))=
VZ(K)(l(z)).
2. A geometric
property
of convex sets
ln this section, K will be a convex body in RN cRN +iRN =CN, i.e., KcRN
is compact, convex, and has non-empty interior. Recall that a real hyperplane
HacRN is a support hyperplane for K at aEoK if aEHa and K lies entirely in one
of the two half-spaces determined by Ha, i.e., if Ha is given by
Ha={XERN
:l(x)=l(a),
lERN*},
.