A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine manifold is given by charts whose transitions functions are affine. I have proved the following, but would be grateful if someone could provide a reference.

Suppose $M$ is a connected affine manifold and $f:M\longrightarrow {\mathbb R}$ is a convex function, which is strictly convex at some point. Given $\epsilon>0$ there is $g:M\longrightarrow {\mathbb R},$ which is smooth, strongly convex and satisfies $|f-g|<\epsilon$.

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    $\begingroup$ Daryl: Maybe you should clarify what you mean by an affine manifold: Some people assume that this is just a manifold equipped with an affine connection, other's assume that the connection is flat. $\endgroup$ – Misha May 11 '14 at 18:22

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