Line segments in a vector space

Let $X$ be a vector space and let $a,b\in X$.

• The closed line segment joining $a$ and $b$ is $[a,b]:=\{\lambda a+(1-\lambda)b:\lambda\in[0,1]\}.$
• The open segment is $(a,b):=\{\lambda a+(1-\lambda)b:\lambda\in(0,1)\}.$
• The half-open segments are $[a,b):=\{\lambda a+(1-\lambda)b:\lambda\in(0,1]\}$ and $(a,b]:=\{\lambda a+(1-\lambda)b:\lambda\in[0,1)\}$.

Context. A set is convex exactly when it contains $[a,b]$ for every $a,b$ in the set.

Examples:

• In $X=\mathbb{R}$, $[a,b]$ is the usual interval between $a$ and $b$.
• In $X=\mathbb{R}^2$, $[a,b]$ is the straight segment in the plane from $a$ to $b$.