Given a convex, open set $K$ in a normed space $X$ such that $0 \in K$, for each $x \in K$, the Minkowski functional is defined as $\rho_K(x) := \text{inf}\{t > 0: \frac{x}{t} \in K\}$. Since $K$ is open, $\exists r >0$ such that $B(0,r) \subseteq K$. The lemma states $\rho_K(x) \leq \|x\| / r$. I have been thinking of this lemma for a while, but still have no clue.
If $0<s<r$ and $y=\frac {sx} {||x||}$ then $||y||=s<r$ so $y \in B(0,r)$. Hence $y \in K$. By definition of $\rho_K$ this gives $\rho_K (x) \leq \frac {||x||} s$ (because $t=||x|| /s$ is in the set $\{t>0 ;x/t \in K$). Now take limit as $s$ increases to $r$.