On uniformly consistent tests

Necessary and sufficient conditions for uniform consistency of sets of alternatives are explored. Hypothesis is simple. Sets of alternatives are bounded convex sets in $\mathbb{L}_p$, $p>1$, with "small" balls deleted. The balls have the center at the point of hypothesis and radii of balls tend to zero as sample size increases. For problem of hypothesis testing on a density, we show that convex sets is uniformly consistent, if and only if, convex set is compact. Similar results are established for signal detection in Gaussian white noise, for linear ill-posed problems and so on.