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Consider the SDP matrices

$$A = \pmatrix{1&0&-1&0\cr 0&1&0&-1\cr -1&0&1&0\cr0&-1&0&1\cr},\ B=\pmatrix{ 1&\sqrt{3}/2 & 1/2&0\cr \sqrt{3}/2&1& \sqrt{3}/2& 1/2\cr 1/2& \sqrt{3}/2&1& \sqrt{3}/2 \cr 0& 1/2& \sqrt{3}/2&1\cr}$$ where with $a=1/2$ $$C = \Phi^{-1}(\Phi(A)/2 + \Phi(B)/2) = \pmatrix{ 1&1/2&-1/2&0\cr1/2&1&1/2&-1/2\cr-1/2&1/2&1&1/2\cr0&-1/2&1 /2&1\cr}$$ is not postive positive semidefinite.

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Consider the SDP matrices

$$A = \pmatrix{1&0&-1&0\cr 0&1&0&-1\cr -1&0&1&0\cr0&-1&0&1\cr},\ B=\pmatrix{ 1&\sqrt{3}/2 & 1/2&0\cr \sqrt{3}/2&1& \sqrt{3}/2& 1/2\cr 1/2& \sqrt{3}/2&1& \sqrt{3}/2 \cr 0& 1/2& \sqrt{3}/2&1\cr}$$ where with $a=1/2$ $$C = \Phi^{-1}(\Phi(A)/2 + \Phi(B)/2) = \pmatrix{ 1&1/2&-1/2&0\cr1/2&1&1/2&-1/2\cr-1/2&1/2&1&1/2\cr0&-1/2&1 /2&1\cr}$$ is not postive semidefinite.