---
tag: 4YZVFGJVSZ
kind: concept
name: Capacity of a set
msc: [31B15, 31C15]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
same_as:
  wikidata: Q5034494
  wikipedia: Capacity of a set
---

## Summary

The Newtonian capacity of a compact set $K \subseteq \R^n$, $n \ge 3$, measures how much electric charge $K$ can hold at unit potential.
It is the least Dirichlet energy of a function that is at least $1$ on $K$ and decays at infinity.
Capacity is the right notion of size in [[potential theory|9PM8Q2E30Q]]: sets of capacity zero are the negligible sets for harmonic functions and for Sobolev functions.
In the plane its role is played by [[logarithmic capacity|DBQXRTC0VN]].

## Explanations

### L1 human

Imagine $K$ as a piece of metal held at voltage $1$, with voltage $0$ far away.
The capacity is the amount of charge that $K$ then carries, and it depends on the shape of $K$ and not only on its volume.
A ball of radius $r$ in $\R^3$ has capacity $4 \pi r$ in the normalisation used here.
A single point and a line segment in $\R^3$ have capacity zero, while a flat disc has positive capacity.
So capacity sees thin sets in a different way from volume.