Golden-Thompson inequality and Lieb's theorem

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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?

  [1]: http://en.wikipedia.org/wiki/Matrix_exponential#Golden.E2.80.93Thompson_inequality