Let $x_1, ..., x_n$ be a set of real numbers. The convex polytope formed from $x_1, ..., x_n$ is given by the set of all $$\sum_i \alpha_i x_i,$$ where $\alpha_i \geq 0$ satisfy $\sum_i \alpha_i =1$.
Suppose instead, however, that we consider the weighted combination $$\sum_i \alpha_i x_i,$$ where $\alpha_i \in [0,1]$. That is, we drop the assumption that $\sum_i \alpha_i$.
Does anyone have any advice/tools for visualising this?
I will assume in a first step that the $x_i$s are vectors of $\mathbb{R^2}$ (as you are mentionning the word polytope). What you are describing is a zonohedron which is defined either as the polygon defined by the so-called Minkowski sum of $n$ line segments (see graphics here) or the projection into $\mathbb{R^2}$ of the hypercube/tesseract of $\mathbb{R^n}$.
Now, as you are in $\mathbb{R^1}$ (the $x_i$s are a set of real numbers), project again the obtained 2D shape onto any axis.