4 problems

Let $NHM_n^m$ be the cone of nonnegative $m$-hemimetrics. Denote by $T_{i_1\dots i_{m+1},i_{m+2}}$ an $(m+1)$-simplex facet and by $N_{i_1\dots i_{m+1}}$ a nonnegativity facet. Let…

Let $HM_n^m$, $NHM_n^m$, and $P_n^m$ be the corresponding cones of $m$-hemimetrics, and let the ridge graph of a cone have the facets as vertices, with adjacency when two facets me…

Let $V_n$ be the ground set, and let $\alpha(S_1,\dots,S_{m+1})$ and $\alpha(T_1,\dots,T_{m+1})$ be two partition $m$-hemimetrics on $V_n$. Two rays are adjacent when they generate…

Let $V_n=\{1,\ldots,n\}$. For $m>1$, consider the cones $P_n^m$, $NHM_n^m$, and $HM_n^m$ of $m$-hemimetrics on $V_n$, together with their faces and symmetries. Symmetry-group conje…