The main contribution of this paper is the proof of the convexity of the omni-directional tethered robot workspace (namely, the set of all tether-length-admissible robot configurations), as well as a set of distance-optimal tethered path planning algorithms that leverage the workspace convexity. The workspace is proven to be topologically a simply-connected subset and geometrically a convex subset of the set of all configurations. As a direct result, the tether-length-admissible optimal path between two configurations is proven exactly the untethered collision-free locally shortest path in the homotopy specified by the concatenation of the tether curve of the given configurations, which can be simply constructed by performing an untethered path shortening process in the 2D environment instead of a path searching process in the pre-calculated workspace. The convexity is an intrinsic property to the tethered robot kinematics, thus has universal impacts on all high-level distance-optimal tethered path planning tasks: The most time-consuming workspace pre-calculation (WP) process is replaced with a goal configuration pre-calculation (GCP) process, and the homotopy-aware path searching process is replaced with untethered path shortening processes. Motivated by the workspace convexity, efficient algorithms to solve the following problems are naturally proposed: (a) The optimal tethered reconfiguration (TR) planning problem is solved by a locally untethered path shortening (UPS) process, (b) The classic optimal tethered path (TP) planning problem (from a starting configuration to a goal location whereby the target tether state is not assigned) is solved by a GCP process and $n$ UPS processes, where $n$ is the number of tether-length-admissible configurations that visit the goal location, (c) The optimal tethered motion to visit a sequence of multiple goal locations, referred to as