  In this paper we build provably near-optimal, in the minimax sense, estimates
of linear forms and, more generally, "$N$-convex functionals" (the simplest
example being the maximum of several fractional-linear functions) of unknown
"signal" known to belong to the union of finitely many convex compact sets from
indirect noisy observations of the signal. Our main assumption is that the
observation scheme in question is good in the sense of A. Goldenshluger, A.
Juditsky, A. Nemirovski, Electr. J. Stat. 9(2) (2015), arXiv:1311.6765, the
simplest example being the Gaussian scheme where the observation is the sum of
linear image of the signal and the standard Gaussian noise. The proposed
estimates, same as upper bounds on their worst-case risks, stem from solutions
to explicit convex optimization problems, making the estimates
"computation-friendly."
