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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 $f\le p|_X$. 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?

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?