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I can prove that the set of symmetric positive-definite matrices with trace $1$ in the space of all symmetric matrices is convex, because $a A + (1-a) B$ is also symmetric positive-definite with trace $1$ when $A, B$ are so and $a\in [0,1]$.

But I have no idea about whether this set is strictly convex and how to prove that.

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    It suffices to note that within the subspace of symmetric matrices, the positive definite matrices form an open set. – 2017-01-04
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    @Omnomnomnom Thanks! I add a constraint that the trace of the matrices are all $1$. What about the new case? – 2017-01-04
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    Same idea. Now, we're considering an open subset of the *affine* space of symmetric matrices with trace $1$. – 2017-01-04
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    @Rahul $A=\begin{bmatrix}0.4&0\\0&0.6\end{bmatrix}, B=\begin{bmatrix}0.6&0\\0&0.4\end{bmatrix}$ while their average is half of the identity matrix. – 2017-01-04
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    @Rahul I'm sorry. We can just append the third diagonal elements $0$ for $A,B$, then they will be positive *semidefinite*. – 2017-01-04
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    @Rahul it is not at all clear what your comment has to do with the question – 2017-01-05
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    @Omnomnomnom: Oops, I was thinking of strict convexity of the set of symmetric positive semidefinite matrices (which is false). – 2017-01-05
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    What does it even mean for a _set_ to be strictly convex? Strict convexity is something you talk about with _functions_. – 2017-01-06
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    @MichaelGrant "Furthermore, a set (in the vector space) is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the interior of the set." from [Convex set](https://en.wikipedia.org/wiki/Convex_set). – 2017-01-06
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    Hmm. So that means it's non-polyhedral (and has no hyperplane faces at all), and has a non-empty interior. I've not personally seen that definition used in practice. I wonder who does. – 2017-01-06
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    So I looked at the Wikipedia history. The term "strictly convex" was added by an anonymous author, that's _all_ they added, and they provided no reference to support the use of the term. I suppose it's definition is reasonably sensible, but still, I've not seen it used, and I've never taught it. I question how useful it is. Do you have a particular reason you need to prove strict convexity? – 2017-01-06
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    Related (but not a duplicate): http://math.stackexchange.com/questions/984963/strictly-convex-set http://math.stackexchange.com/questions/408947/strictly-convex-sets – 2017-01-06
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    The definitions I am seeing involve the boundary points, which are semidefinite. You can't limit your view to the interior points. If that is the case, then the set of positive definite matrices is not strictly convex. – 2017-01-06

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