This paper discusses the concept of the minimal value set. Based on the number of connected components of the minimal value set, the unimodal function (number of connected components = 1) and the multimodal function (number of connected components ≧ 2) are defined. Several properties are derived from the relation between the connected level set and the minimal value set, and the unimodal region of a certain minimal value set is defined. Three theorems concerning the minimal value set are derived, as follows. (1) The set of local minimal points includes the minimal value set. (2) The minimal value set includes the set of minimal points in the narrow sense. (3) The unimodal function includes the quasi-convex function. The probability that the multistart method, where the local optimization technique is applied to the multiple candidate points, finds the global minimal point, as well as the necessary condition for the local optimization technique so that the points on the unimodal region containing the global minimal point converge correctly to the global minimal point, are discussed. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(1): 42–51, 1998