---
tag: WNHXE7FCG4
kind: concept
name: John ellipsoid
msc: [52A27, 52A40, 52A20]
sources:
  - url: https://en.wikipedia.org/wiki/John_ellipsoid
same_as:
  wikidata: Q1699024
  wikipedia: John ellipsoid
---

## Summary

The John ellipsoid of a convex body $K$ in $\R^n$ with nonempty interior is the ellipsoid of largest volume contained in $K$.
It exists and is unique by the [[Löwner–John theorem|TM5SYH4XHY]], and $K$ is contained in its dilate by the factor $n$ about its center.
It gives a canonical affine position of $K$.
It is the starting point of many estimates in convex geometry and the geometry of normed spaces.
Its dual partner is the Löwner ellipsoid, the ellipsoid of least volume containing $K$.

## Explanations

### L1 human

Inside any convex shape one can fit ellipses (or ellipsoids, in space) of many sizes and directions.
Among them there is exactly one of largest area: the John ellipsoid.
For a square it is the inscribed circle, and for a long rectangle it is the inscribed ellipse that touches all four sides at their midpoints.
Every convex shape is squeezed between its John ellipsoid and a copy of it blown up by a factor equal to the dimension.