Set Estimation Under Biconvexity Restrictions

A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments). 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 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 $\theta$ is also given.