Convex set

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Definition

Let $(X,\mathbb{K})$ be a vector space over the field $\mathbb{K}$ which is either the reals, $\mathbb{R}$ or the complex numbers, {{M\mathbb{C} }} and let $C\in\mathcal{P}(X)$ be given. Then we say $C$ is convex if:

• $\forall x,y\in C\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in C]$

Useful notes

1. Notice that $x+t(y-x)\eq (1-t)x+ty$
2. We can also write $[x,y]$ as the (closed) line between $x$ and $y$ by abuse of notation for the notation of a closed interval
   • That is to say: $[x,y]:\eq\{x+t(y-x)\ \vert\ t\in [0,1]\}$