# Metric projection on CAT(0) tangent cone

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.