---
tag: RNS272QBGT
kind: concept
name: Brunn–Minkowski theory
msc: [52A39, 52A40, 52A20]
sources:
  - url: https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem
---

## Summary

Brunn–Minkowski theory is the part of convex geometry that studies [[convex bodies|437FSG3419]] through the interplay of volume and [[Minkowski addition|SXS6D87PHE]].
Its central objects are the [[mixed volumes|QB3Q5FBJ08]], the coefficients of the polynomial that gives the volume of a Minkowski combination of convex bodies.
Its central results are inequalities between them, starting from the [[Brunn–Minkowski theorem|9B02RPN7XD]] and leading to the Alexandrov–Fenchel inequality.
Surface area, mean width, quermassintegrals and the isoperimetric inequality all fit into this framework.

## Explanations

### L1 human

Add two convex shapes by adding each point of one to each point of the other, and ask how the volume of the result depends on the pieces.
The volume of $s K + t L$ turns out to be a polynomial in $s$ and $t$, and its coefficients measure how $K$ and $L$ interact.
For a convex body $K$ and the unit ball $B$, the volume of $K + t B$ is a polynomial in $t$ whose coefficients include the volume and the surface area of $K$.
Brunn–Minkowski theory studies these coefficients and the inequalities between them.