  Optimal estimation and inference for both the minimizer and minimum of a
convex regression function under the white noise and nonparametric regression
models are studied in a nonasymptotic local minimax framework, where the
performance of a procedure is evaluated at individual functions. Fully adaptive
and computationally efficient algorithms are proposed and sharp minimax lower
bounds are given for both the estimation accuracy and expected length of
confidence intervals for the minimizer and minimum.
  The nonasymptotic local minimax framework brings out new phenomena in
simultaneous estimation and inference for the minimizer and minimum. We
establish a novel uncertainty principle that provides a fundamental limit on
how well the minimizer and minimum can be estimated simultaneously for any
convex regression function. A similar result holds for the expected length of
the confidence intervals for the minimizer and minimum.
