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I am a bit confused as to the relationship between the ideas of vector space, affine space, and convex sets in the context of Euclidean space $\mathbb{R}^d$.

As of now, this is how I see it. $\mathbb{R}^d$ is a vector space, and affine spaces are subsets of this vector space that are translates of linear subspaces. Convex sets are subsets of this vector space where any two vectors in the set satisfy the property of convexity.

To what extent is this view correct or incorrect? Here are some of my questions (remember, with regard to $\mathbb{R}^d$ as a vector space): 1) Must convex sets always be subsets of affine spaces? 2) Are affine spaces always subsets of the vector space $\mathbb{R}^d$? 3) Are all affine spaces by definition translates of linear subspaces of $\mathbb{R}^d$, or are there different definitions?

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Yes, affine sets in $\mathbb R^{d}$ are translates of linear subspaces of $\mathbb R^{d}$ and convex sets $S$ are defined by the condition $tx+(1-t)y \in S$ whenever $0\leq t\leq 1$, $x \in S$ and $y \in S$.

1) The entire space $\mathbb R^{d}$ is itself a affine so every convex set is certainly a subset of an affine set.

2) Yes

3) Yes

It should be noted that convex sets and affine sets can also be defined (in the same way) in any vector space.

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  • $\begingroup$ @Murthy I have two follow-up questions. 1) I have also seen affine spaces to be defined as those sets of which are closed under affine combination. How does this square with defining affine spaces as translates of linear subspaces? 2) Wikipedia defines an affine space as "a set $A$ together with a vector space ${\displaystyle {\overrightarrow {A}}}$, and a transitive and free action of the additive group of ${\displaystyle {\overrightarrow {A}}}$ on the set $A$." This definition doesn't seem like it relates to the translate of a linear subspace... $\endgroup$ – Wesley Feb 22 at 14:44

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