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Denote by $\mathbb{P}_{n}$ be the set of positive definite $n \times n$ matrices. In $\mathbb{P}_{n},$ we define an partial order as follow: for $X, Y \in \mathbb{P}_{n},$ we say $X<Y$ if $Y-X$ is also a positive definite matrix. Let $\lambda=2^{1-1 / p}$ with $p>1$ be a real number.
Question: For $0<X<Y<\lambda X,$ do there exist positive definite matrices $A, B$ such that
$$
X=\frac{A+B}{2}, \quad Y=\left(\frac{A^{p}+B^{p}}{2}\right)^{1 / p} ?
$$
Remark: The answer in the case of a scalar is Yes, and $\lambda=2^{1-1 / p}$ comes from this case to ensure that the function
$$
f(a)=\left(\frac{a^{p}+(2 x-a)^{p}}{2}\right)^{1 / p}
$$
is surjective from $[0,2 x]$ to $[x, \lambda x]$

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