The alpha-Embracing Contour

Every notion of depth induces a stratification of the plane in regions of points with the same depth with respect to a given set of points. The boundaries of these regions, also known as depth-contours, are an appropriate tool for data visualization and have already been studied for some depths like Turkey depth [5, 9, 10, 11] and Delaunay depth [3, 8]. The contours also have applications in quality illumination as is the case of good alpha-illumination [2]. The first alpha-depth contour is also known as the alpha-embracing contour. We prove that the first alpha-depth contour has linear size and we give an algorithm to compute it that runs in O(n² log n) time and O(n²)space.