For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the $$L_1$$L1 geodesic diameter in $$O(n^2+h^4)$$O(n2+h4) time and the $$L_1$$L1 geodesic center in $$O((n^4+n^2 h^4)\alpha (n))$$O((n4+n2h4)ź(n)) time, respectively, where $$\alpha (\cdot )$$ź(·) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $$O(n^{7.73})$$O(n7.73) or $$O(n^7(h+\log n))$$O(n7(h+logn)) time, and compute the geodesic center in $$O(n^{11}\log n)$$O(n11logn) time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $$L_1$$L1 shortest paths in polygonal domains.