surfaces in Euclidean space, and their metric properties play an important role in the geometrization conjecture in low-dimensional topology. Hyperbolic convex hulls have also been used as part of the calculation of canonical triangulations of hyperbolic manifolds, and applied to determine the equivalence of knots. See also the section on Brownian motion for the application of convex hulls to this subject, and the section on space curves for their application to the theory of developable surfaces. Statistics In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth. In statistical decision theory, the risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules. Combinatorial optimization In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions. In multi-objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions. Economics In the Arrow–Debreu model of general economic equilibrium, agents are assumed to have convex budget sets and convex preferences. These assumptions of convexity in economics can be used to prove the existence of an equilibrium. When actual economic data is non-convex, it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market. Geometric modeling In geometric modeling, one of the key properties of a Bézier curve is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves. In the geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel,