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Consider the set of $d\times d$ matrices that are positive semidefinite and have unit trace. This is a convex set, $S$. Is it possible to think of a geometric center of this set? The criterion for geometric center is an element $\rho$ such that if $\rho + X\in S$, then $\rho - X\in S$

For $d = 2$, it is the identity matrix. For $d>2$, it is not the identity matrix. Indeed $$\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 &0 \\ 0 & 0 & 0 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 &0 \\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 2/3 & 0 & 0\\ 0 & -1/3 &0 \\ 0 & 0 & -1/3 \end{pmatrix} \in S$$ but $$\frac{1}{3}\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 &0 \\ 0 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 2/3 & 0 & 0\\ 0 & -1/3 &0 \\ 0 & 0 & -1/3 \end{pmatrix} \notin S$$

So is it possible to find such an element for $d>2$ or if not, why not?

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When $d\ge3$, there cannot be any geometric centre. Since the trace of a matrix and the property of being a centre are preserved by unitary similarity, we may consider only the case where $\rho$ is a nonnegative diagonal matrix.

If $\rho=\frac1nI$, then a similar example to yours shows that $\rho$ is not a centre.

If $\rho$ contains two different diagonal entries $a>b$ instead, $\rho$ is also not a centre because $$ \pmatrix{a\\ &b\\ &&c\\ &&&\ddots} +\pmatrix{-\frac{a+b}2\\ &\frac{a+b}2\\ &&0\\ &&&\ddots} =\pmatrix{\frac{a-b}2\\ &\frac{a+3b}2\\ &&c\\ &&&\ddots}\succeq0 $$ but $$ \pmatrix{a\\ &b\\ &&c\\ &&&\ddots} -\pmatrix{-\frac{a+b}2\\ &\frac{a+b}2\\ &&0\\ &&&\ddots} =\pmatrix{\frac{3a+b}2\\ &\frac{b-a}2\\ &&c\\ &&&\ddots}\nsucceq0. $$

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  • $\begingroup$ Thank you, that's a nice proof. However, could you add on a little about whether there exists a better way to define the center? For instance math.stackexchange.com/questions/1260067/… and math.stackexchange.com/questions/1377209/… talk about this "center" of a convex set. Is there an operation that satisfies: If $C \odot X\in S \implies C\odot^{-1} X \in S$, where I have replaced + and - in my original question with some operation $\odot$ and a natural inverse operation $\odot^{-1}$ $\endgroup$ – user1936752 Feb 28 at 11:31
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    $\begingroup$ @user1936752 A "geometric center", as defined in your question, only exists for centrally symmetric sets. The two questions you linked to in your comment describe two other notions of center: the centroid and the analytic center (Boyd & Vandenberghe, sec. 8.5.3) respectively. All of these are different. $\endgroup$ – Rahul Feb 28 at 13:15

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