# Convex set

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## Definition

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

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

## Useful notes

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