Kakutani's theorem (geometry)

Kakutani's theorem is a result in the field of geometry and topology, particularly in the study of multi-valued functions and convex sets. It states that if \( C \) is a non-empty, compact, convex subset of a Euclidean space, then any continuous map from \( C \) into itself that satisfies certain conditions has a fixed point. More specifically, consider a set \( C \) that is compact and convex.