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.

翻译：本文探讨了备择集一致一致性的必要与充分条件。假设为简单假设。备择集是 $\mathbb{L}_p$（$p>1$）中的有界凸集，并删除了“小”球。这些球的中心位于假设点，且其半径随样本量增大而趋于零。对于密度假设检验问题，我们证明：凸集一致一致当且仅当该凸集是紧的。类似结果也在高斯白噪声中的信号检测、线性不适定问题等场景中得到建立。