---
tag: 7GK5417BJF
kind: concept
name: Minkowski sum of polytopes
msc: [52B11, 52A39]
specializes: [SXS6D87PHE]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski_addition
---

## Summary

The Minkowski sum of two convex polytopes $P$ and $Q$ is the set of all sums $p + q$ with $p \in P$ and $q \in Q$.
It is again a [[convex polytope|300K6CB1KH]], and it is the special case of [[Minkowski addition|SXS6D87PHE]] for polytopes.
Its faces are sums of faces of $P$ and $Q$.
Its [[normal fan|XJ83PTS3H0]] is the common refinement of the normal fans of $P$ and $Q$.
Minkowski sums lead to [[zonotopes|Z5SA6A67KF]], to [[mixed volumes|QB3Q5FBJ08]] and to the Newton polytopes of products of polynomials.

## Explanations

### L1 human

Slide a copy of $Q$ so that a chosen point of it visits every point of $P$, and collect everything that $Q$ sweeps out.
The swept region is the Minkowski sum $P + Q$.
For example, the sum of a horizontal unit segment and a vertical unit segment is the unit square.
The sum of a triangle and the same triangle turned upside down is a hexagon.