The well groups were introduced by Edelsbrunner, Morozov, and Patel to measure the robustness of geometric features of a function with respect to perturbations. Roughly speaking, the $r$-th well group measures the number of features that cannot be removed by perturbing the function by at most $r$. The Shrinking Wellness Lemma states that the rank of these groups decreases as $r$ increases. In the generality originally stated, it is wrong. We present a counterexample and give conditions under which the result holds. These conditions are general enough to cover most cases in which the well groups have been applied.

翻译：Edelsbrunner、Morozov和Patel引入健康群组，用于度量函数几何特征相对于扰动的稳健性。粗略而言，第$r$个健康群组度量了在扰动幅度不超过$r$时无法消除的特征数量。收缩健康性引理指出，这些群组的秩随$r$增大而递减。在其最初陈述的一般形式下，该引理并不成立。我们提出了一个反例，并给出了使结果成立的条件。这些条件具有足够的一般性，涵盖了健康群组已被应用的大多数情形。