# Zonohedron

A polyhedron expressible as the vector sum of finitely many segments. Zonohedra in an $n$-dimensional space are sometimes called zonotopes. A zonohedron is a convex polyhedron; the zonohedron itself and all its faces (of all dimensions) have centres of symmetry. A sufficient condition for a convex polyhedron to be a zonohedron is that its two-dimensional faces have centres of symmetry. Any zonohedron is the projection of a cube of sufficiently high dimension. A special role is assigned in the class of centrally-symmetric convex bodies to zonoids — limiting cases of zonohedra; they admit a specific integral representation of the support function and are finite-dimensional sections of the sphere in the Banach space $L_1$.

#### References

[1] | E. Bolker, "A class of convex bodies" Trans. Amer. Math. Soc. , 145 (1969) pp. 323–345 |

[2] | W. Weil, "Kontinuierliche Linearkombination von Strecken" Math. Z. , 148 : 1 (1976) pp. 71–84 |

#### Comments

Zonohedra or zonotopes play an important role in convexity (projection bodies, tiling), analysis (Radon transform, vector-valued measures, subspaces of $L_1$) and stochastic geometry (point processes). Modern surveys are [a1]–[a2].

#### References

[a1] | W. Weil, "Zonoids and related topics" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 296–317 |

[a2] | W. Weil, "Zonoids and generalisations" P.M. Gruber (ed.) J.M. Wills (ed.) , Handbook of convex geometry , North-Holland (1992) |

**How to Cite This Entry:**

Zonohedron.

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