Necessary and sufficient conditions of uniform consistency are explored. A hypothesis is simple. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$, $p >1$ with "small" balls deleted. The "small" 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, for the sets of alternatives, there are uniformly consistent tests for some sequence of radii of the balls, if and only if, convex set is relatively compact. The 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.

翻译：本文探讨了一致最优检验的充分必要条件。假设为简单原假设，备择假设的非参数集是 $\mathbb{L}_p$（$p > 1$）中删除“小”球后的有界凸集。这些小球以原假设点为中心，其半径随样本量增加而趋于零。针对密度假设检验问题，我们证明：对于某些球半径序列，若且唯若凸集是相对紧集时，存在一致最优检验。该结论适用于密度假设检验、高斯白噪声中的信号检测、含随机高斯噪声的线性不适定问题等情形。