Theorem 1. A function $f : R^n \rightarrow R$ is convex if and only if the function $g : R \rightarrow R$ given by $g(t) = f(x + ty)$ is convex (as a univariate function) for all $x$ in domain of $f$ and all $y$ $2$ $R^n$. (The domain of $g$ here is all $t$ for which $x + ty$ is in the domain of $f$.)
I can't proof it