  Geometric quantiles are location parameters which extend classical univariate
quantiles to normed spaces (possibly infinite-dimensional) and which include
the geometric median as a special case. The infinite-dimensional setting is
highly relevant in the modeling and analysis of functional data, as well as for
kernel methods.
  We begin by providing new results on the existence and uniqueness of
geometric quantiles. Estimation is then performed with an approximate
M-estimator and we investigate its large-sample properties in infinite
dimension.
  When the population quantile is not uniquely defined, we leverage the theory
of variational convergence to obtain asymptotic statements on subsequences in
the weak topology. When there is a unique population quantile, we show, under
minimal assumptions, that the estimator is consistent in the norm topology for
a wide range of Banach spaces including every separable uniformly convex space.
  In separable Hilbert spaces, we establish weak Bahadur-Kiefer representations
of the estimator, from which $\sqrt n$-asymptotic normality follows. As a
consequence, we obtain the first central limit theorem valid in a generic
Hilbert space and under minimal assumptions that exactly match those of the
finite-dimensional case.
  Our consistency and asymptotic normality results significantly improve the
state of the art, even for exact geometric medians in Hilbert spaces.
