The boundary and the shape of binary images

In this paper we will consider an unknown binary image, of which the length of the boundary and the area of the image are given. These two values together contain some information about the general shape of the image. We will study two properties of the shape in particular. First, we will prove sharp lower bounds for the size of the largest connected component. Second, we will derive some results about the size of the largest ball containing only ones, both in the case that the connected components are all simply connected and in the general case.