In this thesis we consider the $\alpha$-visibility of a simple polygon P : two points in P are said to be $\alpha$-visible each other if the line segment connecting them lies in P and if the length of the segment is shorter than or equal to $\alpha$. The $\alpha$-kernel of a simple polygon P is a set of points in P that are $\alpha$-visible from all points in P. We present an optimal algorithm for finding the $\alpha$-kernel of a simple polygon. The muimum $\alpha$ such that the $\alpha$-kernel of P is not empty is min-K. We prove that for a convex polygon the min-$\alpha_k$ is the radius of the smallest circle that encloses the polygon. For a simple polygon we show that the min-$\alpha_k$ can be computed in O(n) time by a linear programming technique[15] where n is the number of its vertices.
