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I'm independently studying Boyd & Vandenberghe's Convex Optimization and came across the following statement, which I'm trying to prove.

The affine hull of a set $C$ is the smallest affine set that contains $C$.

where affine hull is defined as

$$\textrm{aff}\,C = \{\theta_1x_1 + \dots + \theta_kx_k \vert x_1, \dots, x_k \in C, \theta_1 + \dots + \theta_k = 1\}$$

and an affine set is defined as a set $S$ where for any $x_1, x_2 \in S$ and $\theta \in \mathbb{R}$, we have $\theta x_1 + (1-\theta)x_2 \in S$.

I was reading through the answer provided on this post which involves proving that the affine hull is an affine set by showing that the affine hull is "closed under affine combination". However, is this sufficient to proving that the affine hull is an affine set?

Specifically, does a set $D$ that is closed under affine combination imply that $D$ is an affine set? I understand that the implication "it can be shown that an affine set is closed under affine combination" is true, but I don't see how the converse is true.

Noah Stebbins
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    What’s your definition of “affine set”? – David M. Feb 05 '19 at 04:35
  • You might want to read about this in Rockafellar's Convex Analysis. I assume you do this to eventually learn about relative interiors. Rockafellar's book remains the standard reference on these topics. – max_zorn Feb 05 '19 at 08:51
  • @DavidM., I updated the question with my definition of affine set. – Noah Stebbins Feb 05 '19 at 13:22
  • Your provided definition seems to me to be the mathematical statement of “closed under affine combination” (i.e. you define an affine set to be a set closed under affine combination) – David M. Feb 05 '19 at 13:28
  • @DavidM., I'm not quite sure I follow you. – Noah Stebbins Feb 05 '19 at 13:44
  • Not to be pedantic, but how would you define a set being closed under affine combination? That is, $S$ is closed under affine combination if ____. – David M. Feb 05 '19 at 13:46
  • I would define it as the following. A set $C$ is closed under affine combination if for $x_1, \dots, x_k \in C$, and $\theta_1 + \dots + \theta_k = 1$, then the point $\theta_1 x_1 + \dots + \theta_k x_k$ also belongs to $C$. – Noah Stebbins Feb 05 '19 at 13:55
  • Right, so if $D$ is closed under affine combination then (what you said). In particular, take $k=2$, with $\theta_1=\theta$ and $\theta_2=1-\theta$. Then $D$ is an affine set. – David M. Feb 05 '19 at 14:05
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    @DavidM., wow can't believe I didn't see that. Thank you! – Noah Stebbins Feb 05 '19 at 14:11

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