# 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.
$$f(x)+exp(−y^2)$$.