Minimax testing of hypotheses on the distribution density for ellipsoids in $l_p$

Let $x_1,\dots,x_N$ be an independent sample with distribution density $f(x)$. A minimax problem of testing a simple hypothesis $f=f_0$ against a complex alternative $f\ne f_\theta$, $\theta\in\Phi_{N,p}^1$, is considered (see Definition in § 1). Asymptotic formulas for error probabilities are obtained which correspond to asymptotic minimax sequences of tests under weaker constraints on the, form of the sets $\Phi_{N,p}^1$ than studied in earlier works.