Computations of volumes in five candidates elections

We describe several analytical results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm

Smooth polytopes with negative Ehrhart coefficients

We present examples of smooth lattice polytopes in dimensions 3 and higher where each coefficient of their Ehrhart polynomials that can potentially be negative is indeed negative. This answers a question by Bruns. We also discuss Berline-Vergne valuations as a useful tool in proving Ehrhart positivity results.Comment: 15 pages, 1 figure, 1 table, accepted by Journal of Combinatorial Theory Ser.

Relative Algebro-Geometric stabilities of Toric Manifolds

In this paper, we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds. The reduction of relative Chow stability on toric manifolds will be investigated by the Hibert-Mumford criterion in two ways. One is to consider the criterion for the maximal torus action and its weight polytope. Then we obtain a reduction by the strategy of Ono \cite{Ono13}, which fits into the relative GIT stability detected by Sz\'ekelyhidi. The other way is to use the criterion for $\mathbb{C}^{\times}$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas \cite{D02, RT07}. In the end, we determine the relative K-stability of all toric Fano threefolds and present a counter-example for relatively $K$-stable manifold, but which is asymptotically relatively Chow unstable.Comment: 24 pages, 2 tables: v2 has minor changes and several added reference