---
tag: K7N565NKCV
kind: concept
name: Supporting hyperplane
msc: [52A20]
sources:
  - url: https://en.wikipedia.org/wiki/Supporting_hyperplane
same_as:
  wikidata: Q1134296
  wikipedia: Supporting hyperplane
---

## Summary

A supporting hyperplane of a set $S$ in $\R^n$ is a hyperplane that has all of $S$ on one side.
It also touches $S$ at a point of the [[boundary|05D5EGMFYV]] of $S$.
By the [[supporting hyperplane theorem|DR9BXFWMQB]], a [[convex set|02AS45T3H9]] has a supporting hyperplane through each of its boundary points.
Supporting hyperplanes link convex sets with linear functions: they describe where a linear function is largest on the set.

## Explanations

### L1 human

Put a convex object on a table.
The table top is a supporting plane: the object lies entirely above it and touches it.
In the plane, a supporting line of a disc is a tangent line, and a supporting line of a square through a corner can be tilted in many ways.
For the square $[0, 1]^2$, the line $x = 1$ is a supporting line touching the right side.