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A numerical invariant  $  \tau = \tau ( E) $
of a compact set  $  E $
in the complex plane that is used in the theory of best approximation.

Let  $  K _ {n} $
be the class of all polynomials

$$ 
p _ {n} ( z)  =  z  ^ {n} + c _ {1} z  ^ {n-} 1 + \dots + c _ {n}  $$

of degree  $  n $, 
and let

$$ 
M ( p _ {n} )  =  \max  \{ {| p _ {n} ( z) | } : {z \in E } \}
,
$$

$$ 
m _ {n}  =  \inf  \{ {M ( p _ {n} ) } : {p _ {n} \in K _ {n} } \} ,\  \tau _ {n}  =  {m _ {n} }  ^ {1/n} .
$$

There exists a polynomial  $  t _ {n} ( z) \in K _ {n} $
for which  $  M ( t _ {n} ) = m _ {n} $; 
it is called the Chebyshev polynomial for  $  E $. 
Moreover, the limit

$$ 
\lim\limits _ {n \rightarrow \infty }  \tau _ {n}  =  \tau
$$

exists, and is called the Chebyshev constant for  $  E $.

Restricting oneself to the class  $  \widetilde{K}  _ {n} $
of all polynomials

$$ 
\widetilde{p}  _ {n} ( z)  =  z  ^ {n} + \dots + \widetilde{c}  _ {n}  $$

all zeros of which lie in  $  E $, 
one obtains corresponding values  $  \widetilde{m}  _ {n} , \widetilde \tau   _ {n} , \widetilde \tau   $
and a polynomial  $  \widetilde{t}  _ {n} ( z) $
for which  $  M ( \widetilde{t}  _ {n} ) = \widetilde{m}  _ {n} $(
it is also called the Chebyshev polynomial).

It is known that  $  \tau = \widetilde \tau   = C ( E) = d $, 
where  $  C ( E) $
is the [[Capacity|capacity]] of the compact set  $  E $, 
and  $  d $
is its [[Transfinite diameter|transfinite diameter]] (cf., for example, [[#References|[1]]]).

The concept of the Chebyshev constant generalizes to compact sets  $  E $
in higher-dimensional Euclidean spaces  $  \mathbf R  ^ {m} $
starting from [[Potential theory|potential theory]]. For a point  $  x \in \mathbf R  ^ {m} $, 
let

$$ 
H ( | x | )  =  \left \{ 
\begin{array}{ll}
 \mathop{\rm ln}  
\frac{1}{| x | }
  & \textrm{ for }  m = 2 ,  \\

\frac{1}{| x |  ^ {m-} 2 }
  & \textrm{ for }  m \geq  3 ,  \\
\end{array}

 \right .$$

be the fundamental solution of the Laplace equation, and for a set  $  ( x _ {j} ) _ {j=} 1  ^ {n} \subset  E $, 
let

$$ 
\sigma _ {n} ( E)  =  \sup \left \{ {\min \left \{ {
\frac{1}{n}
 \sum _ { j= } 1 ^ { n }  H ( | x - x _ {j} | ) } : {x \in E  } \right \} }
: {( x _ {j} ) _ {j=} 1  ^ {n} \subset  E  } \right \} .
$$

Then for  $  m = 2 $
one obtains the relation

$$ 
\tau  =  \widetilde \tau    =  C ( E)  =   \mathop{\rm exp} \left ( - \lim\limits _ {n \rightarrow \infty } \ 
\sigma _ {n} ( E) \right ) ,
$$

and for  $  m \geq  3 $
one obtains (cf. [[#References|[2]]]):

$$ 
\tau  =  C ( E)  =  
\frac{1}{\lim\limits _ {n \rightarrow \infty }  \sigma _ {n} ( E) }
 .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Carleson,   "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Walsh,   "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1956)</TD></TR></table>