Before revision:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
After revision:  Assuming L is the total length of P and \delta is the minimum distance between points on disjoint edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.