A set S ⊂ Z 2 of integer points is digital convex if conv(S) ∩ Z 2 = S, where conv(S) denotes the convex hull of S. In this paper, we consider the following two problems. The first one is to test whether a given set S of n lattice points is digital convex. If the answer to the first problem is positive, then the second problem is to find a polygon P ⊂ Z 2 with minimum number of edges and whose intersection with the lattice P ∩Z 2 is exactly S. We provide linear-time algorithms for these two problems. The algorithm is based on the well-known quickhull algorithm. The time to solve both problems is O(n + h log r), where h = min(| conv(S)|, n 1/3) and r is the diameter of S.