# Pareto-optimal front $F$ in $m$-dimensional space can not have more than $\mathbf{H}_{m-2}(F)$ homology groups

I need to prove that a Pareto-optimal front $$F$$ in $$m$$-dimensional space (i.e. $$m > 1$$) can not have more than $$\mathbf{H}_{m-2}(F)$$ homology groups.

What it simply means that in a 2-dimensional Pareto-optimal front can't have 1 and 2-dimensional holes (due to the dominance relation), but it can have a 0-dimensional persistence homology (i.e. number of connected components). A 3-dimensional Pareto-optimal front can't have 2 and 3-dimensional persistence homology (due to the dominance relation), but it can have a 1-dimensional holes (i.e. loops) and so on.

This is my proof strategy:

In a geometrically realized $$k$$-simplex $$\sigma_k$$, the $$k+1$$ points $$\mathbf{u}_1,\mathbf{u}_2, \ldots ,\mathbf{u}_{k+1} \in \mathbb{R}^k$$ are affinely independent. Where $$\mathbf{u}_i = [f_1, f_2, \ldots, f_m] \in \mathbb{R}^m$$ and $$m > 2$$. Then, the simplex determined by them is the set of points \begin{align} \sigma_k=\left\{\theta_1 \mathbf{u}_1 + \theta_2 \mathbf{u}_2 + \ldots + \theta_{k+1} \mathbf{u}_{k+1} \mid \sum_{i=1}^{k+1} \theta_{i}=1 \text { and } \forall\, i\,\,\theta_{i} \geq 0\right\} \end{align} Let's exclude an arbitrary point $$\mathbf{u}_j$$ from $$\sigma_k$$ and construct a new simplex $$\sigma_{k-1}$$: \begin{align} \sigma_{k-1}=\left\{\theta_1 \mathbf{u}_1 + \theta_2 \mathbf{u}_2 + \ldots + \theta_{j-1} \mathbf{u}_{j-1} + \theta_{j+1} \mathbf{u}_{j+1} + \ldots + \theta_{k} \mathbf{u}_{k} \mid \sum_{i=1}^{k} \theta_{i}=1 \text { and } \forall\, i\,\,\theta_{i} \geq 0\right\} \end{align}

Now what I want to show that there are always a subset of points in $$\sigma_{k-1}$$ that Pareto-dominate $$\mathbf{u}_j$$. If I can do that then I can say in a Pareto-optimal front, the highest dimensional simplex that exists is $$\sigma_{k-1}$$.

Hopefully from there the rest of the proof about $$\mathbf{H}_{m-2}(F)$$ should be easy.

But I can't sort this out. Also not sure if I am even going on a right direction.

Alternative strategy: use the idea from Section 4.7.3 from Boyd's Convex Optimization book. The book talks about the notion of boundary $$\mathrm{bd}$$ of a feasible objective space $$\mathcal{O}$$ . But $$\mathrm{bd}\mathcal{O}$$ is not clearly defined in terms of simplicial complexes.