Let $(Y,d)$ be a complete and separable CAT(0) space, fix $y \in Y$. Then, consider the tanget cone $(T_yY,d_y)$ at $y$, i.e. the metric cone over the space of directions, and denote by $0_y$ the 'tip' of such cone. It is well known that $(T_yY,d_y)$ is CAT(0) space and, for any $C\subset T_yY$ convex and closed, the CAT(0) condition grants the existence of a metric projection $P_C$ assigning to each vector $v \in T_yY$, its projection $v^C= $argmin$_Cd_y(v,\cdot)$.

Define an 'inner product' (cleary in absence of an underlying linear structure) imposing $2\langle v,w\rangle_y := \vert v\vert_y^2 +\vert w\vert_y^2 - d^2_y(v,w)$ for any $v,w \in T_yY$ (here $\vert \cdot\vert_y$ = $d_y(\cdot,0_y)$).

I am interested in the properties of the metric projection $P_H$ in this framework for a specific $H \subset T_yY$. Suppose H is flat region passing through $0_y$ and isometric to $\mathbb{R}^2$ inside $T_yY$. Let $h \in H$, $v \in T_yY$ and consider $v^H$, the metric projection of $v$ onto $H$.

My question: is there any relation between the quantities $\langle h,v\rangle_y$ and $\langle h,v^H\rangle_y$?

The reference I am looking into is 'A course in metric geometry - Burago Burago Ivanov' where it treats a splitting theorem for Hadamard spaces.


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