---
tag: 437FSG3419
kind: concept
name: Convex body
msc: [52A20]
specializes: [02AS45T3H9]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_body
same_as:
  wikidata: Q5166516
  wikipedia: Convex body
---

## Summary

A convex body is a [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]] in $\R^n$ with nonempty [[interior|0RZH6W98HK]].
Convex bodies are the main objects of convex geometry: balls, cubes, simplices and all convex polytopes of full dimension are examples.
Volume, surface area, mixed volumes and the Brunn–Minkowski theory are all developed for convex bodies.

## Explanations

### L1 human

A convex body is a solid, bounded, convex shape that includes its boundary.
It must have some thickness in every direction, so a flat disc lying in space does not count as a convex body in $\R^3$.
A solid ball, a solid cube and a solid tetrahedron are convex bodies in space.
A filled triangle is a convex body in the plane.