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?

  • $\begingroup$ this seems like the convex hull of the $\{ \log (x_i)\}$ which is well defined since the set you wrote only makes sense for $x_i >0$ $\endgroup$ Aug 28 '15 at 9:52
  • $\begingroup$ So then the question is what does the exponential transform a convex hull into... I assume this is only over $\mathbb R$, in which case convex hulls aren't that complicated... do you also what to consider this over $\mathbb C$? $\endgroup$ Aug 28 '15 at 9:54
  • $\begingroup$ I am interested in the case in which all points are in $\mathbb{R}^2$. Edited the question to make it clear. One interesting feature of this "multiplicative hull" is that, while the convex hull of two points is the straight segment connecting them, the multiplicative hull of two points is not a straight segment. This raises questions such as: how does the multiplicative hull of a segment looks like? etc. $\endgroup$ Aug 28 '15 at 10:57
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    $\begingroup$ I hereby pronounce this the log-convex hull. $\endgroup$ Aug 28 '15 at 13:52

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.


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