My problem sheet asks the following:
Let $X$ be a locally convex space generated by the semi-norms $(p_i)_{i\in I}$. Let $A$ be some convex neighborhood of $0$, and $p$ the Minkowski gauge of $p$. Prove that for some $i\in I$, there exists $C>0$ such that $$p\leq C p_i$$
This seems false to me, if there are no other assumptions about the $p_i$. For instance, take $\mathbb R^2$ with the "distance to the $x$-axis" and "distance to the $y$-axis" norms. Then if $A$ is the region between the lines $y = x + 2$ and $y = x - 2$, we have $p(1, 0)>0$ and $p(0, 1)>0$, so $p$ can't be bounded by either semi-norm.