The transfinite diameter of a nonempty compact set 
in the complex
plane is 
, where the 
th diameter is the largest geometric
mean of pairwise distances among 
points in 
. For 
,
![d_n(E)=max_(z_1,...,z_n in E)(product_(1<=j<k<=n)|z_j-z_k|)^(2/[n(n-1)]).](/images/equations/TransfiniteDiameter/NumberedEquation1.svg) |
The points may repeat, so the formula also applies to finite 
. The numbers 
decrease nonstrictly to the logarithmic
capacity of 
(Ransford 1995).
If the connected component of the complement of 
in the Riemann sphere containing the point
at infinity is simply connected and 
, the unique normalized conformal
mapping from the exterior of the unit disk onto
this domain has the Laurent
series
 |
with 
and 
. The coefficient 
is the exterior conformal
radius, and the corresponding Robin constant
is 
(Ransford 1995).
The transfinite diameter also bounds the number of connected components of the open set 
when all the roots
of a monic polynomial 
of polynomial degree 
lie in 
. Ghosh and Ramachandran (2024) proved that if 
, there is an 
depending on 
such that this number is at most 
for all sufficiently large 
. If 
is a connected set and 
, the open set
has exactly one connected component for every
.
At capacity 1, the line segment ![E=[-2,2]](/images/equations/TransfiniteDiameter/Inline28.svg)
lies in no closed disk
of radius 1. The monic
polynomial 
has all its roots in 
, where 
is the Chebyshev
polynomial of the first kind. The corresponding open
set has 
connected components for every 
. This answers the existential reading of problem
1042(i) in the Erdős problems collection
(Ghosh and Ramachandran 2024). It remains open whether every compact
set of capacity 1 that lies in no closed disk
of radius 1 permits 
connected components
for infinitely many 
.
