On estimation of biconvex sets
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A set in the Euclidean plane is said to be θ-biconvex, for some θ∈[0,π/2), when all its sections along the directions θ and θ+π/2 are convex sets in the real line. Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of `rectilinear convexity', in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the `biconvex hull' of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of `hull' for a given set A (understood as the `minimal' set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of `biconvex hull' (often called `rectilinear convex hull'): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation (with respect to the Hausdorff metric) of a biconvex set S and and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate (from a sample of points on S) the biconvexity angle θ is also given.
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