Necessary and sufficient conditions of uniform consistency are explored. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$ 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 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：本文探讨了一致一致性的充分必要条件。非参数备择集合定义为$\mathbb{L}_p$中有界凸集并剔除“小”球邻域。这些“小”球的中心位于假设点处，其半径随样本量增大而趋于零。针对密度假设检验问题，我们证明：当且仅当凸集为紧集时，存在关于某球半径序列的一致一致检验。该结论适用于密度假设检验、高斯白噪声中的信号检测、随机高斯噪声下的线性不适定问题等场景。