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arXiv · 2601.03275

Sufficient Conditions for the Shrinking Wellness Lemma

Clemens Bannwart
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Abstract

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.

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Clemens Bannwart. 2025-12-25. Sufficient Conditions for the Shrinking Wellness Lemma. https://arxiv.org/abs/2601.03275

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