At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: *Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\to \mathbb R$ linear with $f\le p|\_L$. Then there is a linear $F:X\to\mathbb R$ with $F\le p$ and $F|\_L=f$*.


However the theorem is true if the majorant $p$ is merely convex. This version has a very similar proof as the classical statement and several advantages. For instance, there is no need to introduce the new notion of sublinearity and the result is even interesting for $X=\mathbb R$.


The only reference I know is the book of Barbu und
Precupanu *Convexity and Optimization in Banach Spaces*.


Two questions:


1. Who first observed that sublinearity can be replaced by convexity?
2. Is there any (e.g. pedagocial) reason to prefer the sublinear version?



Hahn-Banach theorem with convex majorant
If $p$ is convex, then $P(x)=\inf\_{t>0}t^{-1}p(tx)$ is sublinear, isn't it? Also, if a *linear* functional is dominated by $p$, it is also dominated by $P$. Finally, $P\le p$. So there is no non-trivial gain in generality whatsoever *unless* you start talking about extending non-linear functionals but then you should restate the question accordingly.


