Definition 5

Tag 07CA, permanent. Cite it as [Bourbaki, Tag 07CA].

Topological Vector Spaces, II, p. 16, § 2. Convex sets

Let X be a convex subset of the affine space E. A real-valued finite function, defined over X is convex (resp. strictly convex) if for any two distinct points x, y of X and any real number $\lambda, 0 < \lambda < 1$ , we have:

(1)

f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y)

(resp.

(2)

f(\lambda x + (1 - \lambda) y) < \lambda f(x) + (1 - \lambda) f(y)).

When $E = \mathbf{R}$ , this definition of convex function is the same as that in FVR, I, p. 32. Further, f is convex (resp. strictly convex) in X if, and only if, for every affine line $D \subset E$ , the restriction of f to $X \cap D$ is convex (resp. strictly convex) in $X \cap D$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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