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We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.
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In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $Mto B$, $B$ is connected. For $varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_varepsilon$. The main result says that $lim_{varepsilonto +0} V_i(M_varepsilon)=chi(Z) V_i(B)$, where $chi(Z)$ is the Euler characteristic of a fiber of the submersion. This result is consistent with more general open conjectures on convergence of intrinsic volumes formulated previously by the author.
We describe several experimental results obtained in four candidates social choice elections. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.
In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in $X$ on the equator and point-hyperplane frameworks with the vertices in $X$ representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.
Let $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$ contains an imaginary number. We would like to classify the largest possible $s$-codes for given dimension $d$. In this paper, we consider the problem for the case $s=3$. Roy and Suda (2014) gave a certain upper bound for the cardinalities of $3$-codes. A $3$-code $X$ is said to be tight if $X$ attains the bound. We show that there exists no tight $3$-code except for dimensions $1$, $2$. Moreover we make an algorithm to classify the largest $3$-codes by considering representations of oriented graphs. By this algorithm, the largest $3$-codes are classified for dimensions $1$, $2$, $3$ with a current computer.
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullens polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.
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