I'm trying to prove that given a subset $C \subseteq (X, \parallel \cdot \parallel )$ normed vector space, $0 \in C$, and its Minkowski functional $p_C$, then it holds
$C$ convex iff $p_C$ satisties the triangle inequality , $p_C(x+y) \leq p_C(x) + p_C(y)$.
I have "$ \Rightarrow$".
"$ \Leftarrow$" holds if I prove that $C= \{ x \in X : p_C(x) \leq 1 \}$ (and I need to prove this equality!). "$\subseteq$" is trivial, but I don't know how to prove "$\supseteq$". All bibliography I have read proves this equalily knowing that $C$ is convex, but I can only use that $p_C$ satisties the triangle inequality.
So, ¿how to prove that if $x \in X$ verifies $p_C(x) \leq 1$ then $x \in C$?
Thanks in advance.