Let $\Lambda$ be a finite set. Let $\Lambda^{(2)}$ be the set of unordered pairs $\{x,y\}\subset\Lambda$ with $x\neq y$. I will use ${\rm Part}(\Lambda)$ to denote the set of set partitions of $\Lambda$. The polyhedron $K$ I am interested in lives in the vector space $$ V=\mathbb{R}^{\Lambda^{(2)}}\ . $$ A vector $v$ is therefore a collection of real numbers $v=(v_{\{x,y\}})_{\{x,y\}\in\Lambda^{(2)}}$ indexed by unordered pairs of elements in $\Lambda$, i.e., edges of the complete graph with vertex set $\Lambda$. To a partition $\Pi\in{\rm Part}(\Lambda)$, let me associate a vector $v_{\Pi}$ in $V$ whose components $v_{\Pi,\{x,y\}}$ are, by definition, equal to $1$ if $x,y$ are in the same block of partition $\Pi$, and equal to $0$ otherwise.

Definition: The set $K$ is the convex hull of all the $v_{\Pi}$, with $\Pi\in{\rm Part}(\Lambda)$.

An alternate description using real symmetric matrices is as follows. Let $\widetilde{V}$ be the vector space of real symmetric matrices $M=(M(x,y))_{x,y\in\Lambda}$ with rows and columns indexed by $\Lambda$ (e.g., $\Lambda=[n]:=\{1,2,\ldots,n\}$, for more comfort). To a $v\in V$ one can associate a matrix $M\in\widetilde{V}$ by letting $M(x,y)=v_{\{x,y\}}$ if $x\neq y$, and letting $M(x,x)=1$ for all $x\in\Lambda$. Let $\widetilde{K}\subset\widetilde{V}$ be the image of $K$ by the map just defined. It is not hard to see that the symmetric matrices $M$ in $\widetilde{K}$ are characterized by the following properties.

1. For all $x,y\in\Lambda$, we have $0\le M(x,y)\le 1$.
2. For all $x\in\Lambda$, we have $M(x,x)=1$.
3. For all $x,y,z\in\Lambda$, we have $M(x,z)\ge\min(M(x,y),M(y,z))$.

My coauthor Greg W. Anderson, for the unpublished paper "Counting colored planar maps free-probabilistically" (see Section 2.1.1), called these co-ultrametrics because, modulo reversal of inequalities, this is reminiscent of the definition of an ultrametric distance. I love this name very much, but I don't know if there is a standard name already.

My question: Did $K$, or its avatar $\widetilde{K}$, appear in the literature, and does it have a name like associahedron, permutohedron, ...hedron?

My motivation: I am currently teaching a course on mathematical quantum field theory "for undergrads" and will soon need to talk about $K$. I would like to use the most appropriate terminology.

Updated: My first post had a mistake because I confused in my mind two different but related sets. Hopefully the description below is correct now.

Let $\Lambda$ be a finite set. Let $\Lambda^{(2)}$ be the set of unordered pairs $\{x,y\}\subset\Lambda$ with $x\neq y$. I will use ${\rm Part}(\Lambda)$ to denote the set of set partitions of $\Lambda$. The polyhedron $K_1$ I am interested in lives in the vector space $$ V=\mathbb{R}^{\Lambda^{(2)}}\ . $$ A vector $v$ is therefore a collection of real numbers $v=(v_{\{x,y\}})_{\{x,y\}\in\Lambda^{(2)}}$ indexed by unordered pairs of elements in $\Lambda$, i.e., edges of the complete graph with vertex set $\Lambda$. To a partition $\Pi\in{\rm Part}(\Lambda)$, let me associate a vector $v_{\Pi}$ in $V$ whose components $v_{\Pi,\{x,y\}}$ are, by definition, equal to $1$ if $x,y$ are in the same block of partition $\Pi$, and equal to $0$ otherwise.

Definition: The set $K_1$ is the convex hull of all the $v_{\Pi}$, with $\Pi\in{\rm Part}(\Lambda)$.

I am also interested in the subset $K_2\subset K_1$ which is defined as follows. If $\Pi_1,\Pi_2$ are two partitions, we write $\Pi_1\preccurlyeq\Pi_2$ if all blocks of $\Pi_1$ are contained in blocks of $P_2$, i.e., $\Pi_1$ refines $\Pi_2$. Let $K_2$ be the set of all convex combinations $$ \alpha_1\ v_{\Pi_1}+\alpha_2\ v_{\Pi_2}+\cdots+\alpha_{\ell} \ v_{\Pi_{\ell}} $$ where the $\alpha$'s are nonnegative and add up to $1$, and where $$ \Pi_1\preccurlyeq\Pi_2\preccurlyeq\cdots\preccurlyeq\Pi_{\ell} $$ is a chain in the poset $({\rm Part}(\Lambda),\preccurlyeq)$.

An alternate description using real symmetric matrices is as follows. Let $\widetilde{V}$ be the vector space of real symmetric matrices $M=(M(x,y))_{x,y\in\Lambda}$ with rows and columns indexed by $\Lambda$ (e.g., $\Lambda=[n]:=\{1,2,\ldots,n\}$, for more comfort). To a $v\in V$ one can associate a matrix $M\in\widetilde{V}$ by letting $M(x,y)=v_{\{x,y\}}$ if $x\neq y$, and letting $M(x,x)=1$ for all $x\in\Lambda$. Let $\widetilde{K}_2\subset\widetilde{V}$ be the image of $K_2$ by the map just defined. It is not hard to see that the symmetric matrices $M$ in $\widetilde{K}_2$ are characterized by the following properties.

1. For all $x,y\in\Lambda$, we have $0\le M(x,y)\le 1$.
2. For all $x\in\Lambda$, we have $M(x,x)=1$.
3. For all $x,y,z\in\Lambda$, we have $M(x,z)\ge\min(M(x,y),M(y,z))$.

My coauthor Greg W. Anderson, for the unpublished paper "Counting colored planar maps free-probabilistically" (see Section 2.1.1), called these co-ultrametrics because, modulo reversal of inequalities, this is reminiscent of the definition of an ultrametric distance. I love this name very much, but I don't know if there is a standard name already.

My question: Did $K_1$, appear in the literature, and does it have a name like associahedron, permutohedron, ...hedron? Also, is there a name for $K_2$ or its avatar $\widetilde{K}_2$?

My motivation: I am currently teaching a course on mathematical quantum field theory "for undergrads" and will soon need to talk about these sets. I would like to use the most appropriate terminology.