The convex hull of a finite set of points, $(x_i,y_i) \in \mathbb{R_+}^2$ ($i=1,...,n$), is defined as:
$$\left\{(\sum_{i=1}^{n} \alpha_i x_i,\sum_{i=1}^{n} \alpha_i y_i) \mathrel{\Bigg|} (\forall i: \alpha_i\ge 0)\wedge \sum_{i=1}^{n} \alpha_i=1 \right\}.$$
Define the following set:
$$\left\{(\prod_{i=1}^{n} x_i^{\alpha_i},\prod_{i=1}^{n} y_i^{\alpha_i}) \mathrel{\Bigg|} (\forall i: \alpha_i\ge 0)\wedge \sum_{i=1}^{n} \alpha_i=1 \right\}.$$
Does this set have a name? (maybe "multiplicative hull"?) Has it been studied or used anywhere?
The term "logarithmic convex hull" is in use for this object, because it can be obtained by convexifying the image of a domain under the logarithm map, and then coming back.
It comes up in complex analysis in several variables, due to the following fact: a domain $D\subset\mathbb{C}^n$ is a region of convergence of some power series (centered at $0$) if and only if $D$ is a logarithmically convex Reinhardt domain.