---
tag: VN4M72SQW5
kind: concept
name: Exposed point
msc: [52A20, 46A55]
specializes: [3MSGFJM2NX]
sources:
  - url: https://en.wikipedia.org/wiki/Exposed_point
same_as:
  wikidata: Q21072692
  wikipedia: Exposed point
---

## Summary

An exposed point of a set $C$ is a point at which some linear function attains its maximum over $C$ and attains it nowhere else.
Geometrically, some [[supporting hyperplane|K7N565NKCV]] touches $C$ at that point only.
Every exposed point is an [[extreme point|3MSGFJM2NX]], but not conversely.
Exposed points are the solutions found by linear optimisation over a convex set when the optimum is unique.

## Explanations

### L1 human

A point of a shape is exposed if you can lay a flat wall against the shape so that it touches the shape at that point and nowhere else.
The corners of a square are exposed: a wall placed diagonally at a corner touches only that corner.
Every point of a circle is an exposed point of the disc.
Now take a stadium, a rectangle with two half-discs glued on its short sides.
The points where a straight side meets a round end are extreme, but every supporting line there contains the whole straight side, so they are not exposed.