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We prove that the set of orthogonal projections on a Hilbert space equippedwith the length metric is $\frac\pi2$-geodesic. As an application, we considerthe problem of variation of spectral subspaces for bounded linear self-adjointoperators and obtain a new estimate on the norm of the difference of twospectral projections associated with isolated parts of the spectrum of theperturbed and unpertubed operators, respectively. In particular, recent resultsby Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No.1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476] are sharpened.

Metric properties of the set of orthogonal projections and their applications to operator perturbation theory