  In this paper, we study the local variational geometry of the optimal
solution set of the trust region subproblem (TRS), which minimizes a general,
possibly nonconvex, quadratic function over the unit ball. Specifically, we
demonstrate that a H\"olderian error bound holds globally for the TRS with
modulus 1/4 and the Kurdyka-{\L}ojasiewicz (KL) inequality holds locally for
the TRS with a KL exponent 3/4 at any optimal solution. We further prove that
unless in a special case, the H\"olderian error bound modulus, as well as the
KL exponent, is 1/2. Finally, based on the obtained KL property, we further
show that the projected gradient methods studied in [A. Beck and Y. Vaisbourd,
SIAM J. Optim., 28 (2018), pp. 1951--1967] for solving the TRS achieve a
sublinear or even linear rate of convergence.
