# On the Hahn-Banach Theorem

I just started a self-study of functional Analysis from "Functional analysis, Sobolev spaces and partial differential equations" by Haim Brezis. I struggle to understand the underlying assumptions for the proof of Hahn-Banach theorem on the extension of function. In brief, the theorem is written as follows

Let $$p : E \rightarrow \mathbb{R}$$ be a Minkowski function.

Let $$G \subset E$$, $$g : G \rightarrow \mathbb{R}$$ such that $$g \leq p$$ is a linear function.

Then there exists a linear function $$f$$ such that $$f=g, \forall x \in G$$ and $$f \leq p, \forall x \in E$$.

The definition of a Minkowski is

$$1. p(\lambda x) = \lambda p(x) \forall \lambda > 0$$,

$$2. p(x+y) \leq p(x)+p(y)$$.

My question is why we need the first strict condition for this theorem. Why any convex function does not work with the condition $$p(\lambda x) \leq \lambda p(x), \forall \lambda \geq 1$$.

First, for the case that $$p(0)=0$$ holds, one can show that your conditions on $$p$$ imply that $$p$$ is a Minkowski function.
But (perhaps) more important is that Hahn-Banach only compares the function $$p$$ to a linear function ($$f$$ or $$g$$). For comparing a function $$p$$ with a linear function in this context, it suffices to only consider the worst'' values of $$p$$ along each ray $$\{t x : t>0\}$$.
To formalize this, suppose $$p$$ is convex. We define $$q$$ via $$q(x):=\inf \{t p(x/t) : t>0 \}$$. Then one can show that $$q$$ is a Minkowski function and that $$g \leq p \iff g \leq q$$ holds for all linear functions $$g$$ which are defined on a linear subspace (I will not give the detailed proof here for these claims).
From this one can conclude that the Hahn-Banach theorem also holds for convex functions $$p$$, and not just Minkowski functions.