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Register a new userCombinatorial properties of ultrametrics and generalized ultrametrics
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O. Dovgoshey
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Let $X$, $Y$ be sets and let $Phi$, $Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $Phi$ and $Psi$ are combinatorially similar if there are bijections $f colon Phi(X^2) to Psi(Y^{2})$ and $g colon Y to X$ such that $Psi(x, y) = f(Phi(g(x), g(y)))$ for all $x$, $y in Y$. Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.
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Let $X$, $Y$ be sets and let $Phi$, $Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $Phi$ is combinatorially similar to $Psi$ if there are bijections $f colon Phi(X^2) to Psi(Y^{2})$ and $g colon Y to X$ such that $Psi(x, y) = f(Phi(g(x), g(y)))$ for all $x$, $y in Y$. It is shown that the semigroups of binary relations generated by sets ${Phi^{-1}(a) colon a in Phi(X^{2})}$ and ${Psi^{-1}(b) colon b in Psi(Y^{2})}$ are isomorphic for combinatorially similar $Phi$ and $Psi$. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by ${d^{-1}(r) colon r in d(X^{2})}$ is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $d colon X^{2} to mathbb{R}$.
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.
Monskys celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monskys polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.
We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.
We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.
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