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.



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