Given a convex cone $\mathcal{K}\subseteq \mathbb{R}^n$, we can define a partial order $\leq_\mathcal{K}$on $ \mathbb{R}^n$ by setting $$x\leq_\mathcal{K} y \Leftrightarrow y-x\in \mathcal{K}.$$ For example, using $\mathcal{K}=\mathbb{R}_+^n$ we get the partial order $\leq$ where $x\leq y $ iff. $x_i\leq y_i$ for all $i\in [n]$.
Now assume you have given a set $S\subseteq \mathbb{R}^n$, then $l$ is called a lowerbound to $S$ w.r.t. to $\leq_\mathcal{K}$ when $$l\leq_\mathcal{K} s \quad \forall s\in S,$$ where $u$, the upperbound to $S$ w.r.t. to $\leq_\mathcal{K}$, is defined analogously.
$l$ is called the infimum of $S$ w.r.t. to $\leq_\mathcal{K}$ if $l$ is a lowerbound of $S$ w.r.t. to $\leq_\mathcal{K}$ and if for every other lowerbound $l'$ of $S$ w.r.t. $\leq_\mathcal{K}$ we have $l'\leq_\mathcal{K} l$. The supremum of $S$ w.r.t. to $\leq_\mathcal{K}$ can be defined analogously. We also write $\inf_S(\mathcal{K})$ and $\sup_S(\mathcal{K})$, since by antisymmetry of the partial order, infimum and supremum w.r.t. $\leq_\mathcal{K}$ are unique if they exist.
For example, when $\mathcal{K}=\mathbb{R}^n_+$, the infimum and supremum of a set $S$ is given compnentwise by $$[\inf_S(\mathbb{R}^n_+)]_i=\min_{s\in S}s_i, \quad [\sup_S(\mathbb{R}^n_+)]_i=\max_{s\in S}s_i .$$ In particular, $S$ is contained in a box given by $(\inf_S(\mathbb{R}^n_+)+\mathbb{R}^n_+) \cap (\sup_S(\mathbb{R}^n_+)-\mathbb{R}^n_+)$.
I'm interested in these things:
- Existence and construction/explicit formulas of $\inf_S(\mathcal{S}^n_+)$, $\sup_S(\mathcal{S}^n_+)$, where $\mathcal{S}^n_+$ is the cone of positive semidefinite matrices of dimension $n$
- Any bounds on $\|\inf_S(\mathcal{K})\|$ and $\|\sup_S(\mathcal{K})\|$ in terms of $diam_{\|.\|}(S)$, where $diam_{\|.\|}(S)=\max_{s_1,s_2\in S} \|s_1-s_2\|$. The norm $\|.\|$ is left open, but I would be interested in norms that admit tight bounds.
- Intuition for the case of $\mathcal{S}^n_+$
Is there any literature on this? I know that $\mathbb{R}^n_+$ is a lattice, but that's about everything I know.