Minimax hypotheses testing for nondegenerate loss functions and extreme convex problems

We study some class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for some class of loss functions. Under rather general assumption we obtain the exact asymptotics (of Gaussian type) for minimax error probabolities and the structure of asymptotically minimax tests.
The methods are based on a reduction of the problems under consideration to extremal problems of minimization of some Hilbert norm on convex sets of sequenses of probability measures on the real line. These extremal problems are investigated in [5] for alternating type of $l_q$-ellipsoids with $l_p$-balls removed. Bibl. 16 titles.