No Valleys, Just One Basin

A function $f$ is convex if for all $x, y$ in its domain and $\lambda$ in $[0,1]$ , $f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda) f(y)$ . Why does this property guarantee that any local minimum is also a global minimum?

Convexity only applies to linear functions, which trivially have one minimum

Convexity means the function's graph lies below (or on) any chord connecting two points, which rules out separate local dips that are not the overall minimum

Convexity guarantees the function has a positive-definite Hessian everywhere, which only affects the step size, not the number of minima

Convex functions are always monotonically decreasing until they reach the minimum