# Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are:

1. It mentions that Lieb's thoerem "accomplishes in a way" what is left undone by the Golden-Thompson inequality (which cannot be extended to three matrices). I'm not seeing the connection.
2. It mentions the "cone of positive matrices". What is the meaning of the term "cone" here?

2. From Wikipedia (Convex Cone): A subset $C$ of a vector space $V$ is a convex cone if $αx + βy$ belongs to $C$, for any positive scalars $α$, $β$, and any $x, y$ in C. The set of positive-definite matrices, as a subset of the vector space of n-by-n matrices, is a cone by this definition.