Convex targets make any two maps homotopic by straight-line interpolation.Suggest a slogan
Let \(X\) be a topological space and \(C\subseteq\RR^n\) a convex subset. Any two continuous maps \(f, g\colon X\to C\) are homotopic, via the linear homotopy \(F(x,t) = (1-t)f(x) + t\,g(x)\), which lies in \(C\) by convexity. For \(X = [0,1]\) and paths with common endpoints, the linear homotopy is a homotopy rel endpoints [@Hat02].