On uniform consistency of nonparametric tests

For problem of hypothesis testing on a density we explore condition of existence of uniformly consistent tests. Hypothesis is simple. Sets of alternatives is convex sets in $\mathbb{L}_p$, $p>1$, with deleted balls. Hypothesis is center of balls. We show that, there is sequence of radii of the balls tending to zero as sample size increases such that the sets of alternatives are uniformly consistent, if and only if convex set is compact. Similar results are established for problem of hypothesis testing on a density, for signal detection in Gaussian white noise, for linear ill-posed problems with random Gaussian noise and so on.