---
tag: W3NVXBEMRQ
kind: concept
name: Robin constant
msc: [31A15, 30C85]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_radius
---

## Summary

The Robin constant of a compact set $K$ in the plane is the least logarithmic energy of a unit charge placed on $K$.
It is a number in $(-\infty, \infty]$, and it equals $\infty$ exactly when every unit charge on $K$ has infinite energy.
The [[logarithmic capacity|DBQXRTC0VN]] of $K$ is $e^{-V}$, where $V$ is the Robin constant.
The Robin constant also appears in the expansion at infinity of the Green function of the outside of $K$.

## Explanations

### L1 human

Spread a unit charge on a plane conductor $K$ in the cheapest way, with the logarithmic law of repulsion.
The energy of this best arrangement is the Robin constant.
A big set gives the charge room to spread, so its Robin constant is small, possibly negative.
A single point gives no room at all, and its Robin constant is $+\infty$.
For a disc of radius $r$ the Robin constant is $-\log r$.