unimodal category

We say that a continuous function is unimodal if all the upper excursion sets
\(e_f(c):=\{f≥c\}\) are contractible. For a nonnegative function
\(f:\mathbb{R}^d\to\mathbb{R}_+\) with compact support its unimodal category UCat(f) as the least number \(u\) of summands in a decomposition \(f=\sum_{i=1}^u f_i\), where all \(f_i\) are unimodal. Unimodal category is a “soft” version of Lyusternik-Schnirelman category, hence the name. We know how to compute UCat only in one-dimensional situation. In other dimensions, even the simplest questions seem hard, for example:

Find the unimodal category of the suspension of univariate Morse functions, i.e.


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