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Definition of a convex set. A set S in R^n is said to be convex if for each x1, x2 ∈ S, the line segment λx1 + (1-λ)x2 for λ ∈ (0,1) belongs to S. This says that all points on a line connecting two points in the set are in the set. My question is how can we define convexity in one dimension ?

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In $\mathbb R$, the definition is the same and you can prove that it is equivalent to connexity, or simpler : $$A\subseteq\mathbb R\text{ is convex }\iff A\text{ is an interval}.$$

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  • $\begingroup$ Hm... what does that look like geometrically? $\endgroup$ – Simply Beautiful Art Jul 27 '16 at 12:56
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    $\begingroup$ @SimpleArt: Convexity says "the line segment between any two points is contained in the set". Since $\Bbb{R}$ is a line, a convex subset of $\Bbb{R}$ is itself a line segment, and therefore is an interval. $\endgroup$ – Will R Jul 27 '16 at 13:13
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The formulas in your question(as edited) work perfectly well in $\mathbb{R}$. The collection of convex sets is the collection of intervals.

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  • $\begingroup$ I mean i.e: points x1 and x2 are given at least from two dimensional coordinates. I want this to reduce in just one dimension coordinate and to see how the equation takes form? $\endgroup$ – ACherkessi Jul 27 '16 at 11:56
  • $\begingroup$ Your formula works for ${\mathbb{R}}^{n}$. Set $n = 1$. $\endgroup$ – Jay Jul 27 '16 at 12:07

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