zondiameter

Zondiameter is a term used in geometry to denote the maximum distance between any two points in a zonotope, a convex polytope obtained as the Minkowski sum of finitely many line segments. The name blends zonotope and diameter and is encountered in discussions of distance properties of zonotopes. It is not a universally standardized term, but researchers and educators may use it to refer to the largest separation inside a zonotope.

Definition and interpretation: Let Z be a zonotope generated by vectors v1, v2, ..., vm in Euclidean space, written as Z = sum_i [0, v_i]. The zondiameter D is the greatest Euclidean distance between any two points in Z. A convenient equivalent formulation uses directions: D = max_{||u|| = 1} sum_i |u · v_i|, where u is a unit vector and "·" is the dot product. This follows from the zonotope’s support function and the fact that the diameter equals the maximum width in some direction.

Special cases and examples: In two dimensions, if Z is a parallelogram generated by two vectors v1 and v2, the zondiameter often corresponds to the length of v1 + v2, provided the orientation yields positive projections along a common direction. In more complex zonotopes with many generators, the zondiameter reflects the combined length of the generators when aligned by an optimal direction u.

Computing and properties: The zondiameter can be computed by optimizing the function f(u) = sum_i |u · v_i| over unit vectors u. This optimization is convex and finite-dimensional, allowing grid search, directional sampling, or convex optimization techniques in practice. The concept aligns with other distance measures in convex geometry and helps characterize the overall extent of zonotopes.

Applications: Zondiameters appear in computational geometry, collision detection, shape bounding, and high-dimensional data representation, where understanding the maximal extent of a zonotope informs algorithms and analyses.

See also: Zonotope, Diameter (geometry), Minkowski sum, Convex geometry.