---
tag: ZMZZ2QRFTZ
kind: concept
name: Zonoid
msc: [52A21, 52A20]
specializes: [437FSG3419]
sources:
  - url: https://en.wikipedia.org/wiki/Zonoid
same_as:
  wikidata: Q131417845
  wikipedia: Zonoid
---

## Summary

A zonoid is a [[convex body|437FSG3419]] that is a limit, in the [[Hausdorff distance|F79WYYC07C]], of [[zonotopes|Z5SA6A67KF]], that is, of finite Minkowski sums of line segments.
Zonoids are exactly the convex bodies whose support functions are integrals of the functions $x \mapsto \abs{x \cdot v}$, up to a translation.
They appear in the geometry of finite-dimensional normed spaces and in the study of projections of convex bodies.
By Lyapunov's convexity theorem, they also arise as the ranges of nonatomic vector-valued measures.

## Explanations

### L1 human

Adding line segments in the sense of Minkowski gives polytopes with many parallel edges, such as parallelograms, hexagons with parallel opposite sides, or cubes.
A zonoid is what one gets by adding more and more, ever shorter segments, and passing to the limit.
The round disc is a zonoid: it is the limit of regular polygons with $2m$ sides, each of which is a sum of $m$ segments.
Every zonoid is centrally symmetric, so a triangle is never a zonoid.