This paper explores the design problem of consensus algorithms in a class of convex geometric metric spaces. Using the techniques of convex analysis and possibility analysis, a simple assumption for designing consensus algorithms in a strict max-convex space is proposed, under which all dynamical points in a system achieve consensus asymptotically if and only if their associated interaction graph uniformly contains at least one directed spanning tree. Three efficient consensus algorithms under the assumption are presented, and their applications are demonstrated together with efficiency studies.