arXiv subjects
Publications and source records attributed to Tom Baumbach.
We study rational generalized barycentric coordinates (rational GBCs) on polytopes. We recall their essential theory and clarify some statements from the literature. We study polytopes that admit GBCs with low degree denominators (and hence, low degree (affine) adjoint polynomials). We give a full classification of polytopes that admit polynomial barycentric coordinates (i.e. constant adjoints).
For a fixed set $U=(u_1,\ldots,u_m)$ of outer unit normals, we consider the cone-volume map which associates to a vector of support numbers $b\in\mathbb R^m_{> 0}$ the cone-volume vector $γ\in\mathbb R^m$ of the polytope \[ P(U,b)=\{x\in\mathbb{R}^n:U^\top x\leq b\}. \] We study its fibers $S(U,γ)$ over prescribed cone-volume vectors $γ$, with particular emphasis on finiteness and non-uniqueness. We give characterizations of the irreducibility of $U$ in terms of properties of the cone-volume map and its solution sets. In particular, for irreducible $U$, we show that $S(U,γ)$ is finite for all $γ$ outside a set of measure zero in the cone-volume cone, and we give an explicit upper bound for its cardinality. In the planar case, we prove that $S(U,γ)$ has semialgebraic dimension at most one for arbitrary $U$. Moreover, for every $U\in\mathcal{U}(2,m)$ with $m\geq 4$, there exists a cone-volume vector $γ$ for which $S(U,γ)$ contains at least two distinct elements. As a consequence, there exist two distinct smooth planar convex bodies with the same cone-volume measure. Finally, for $U$ in general position, we show that the number of solutions is odd for every cone-volume vector outside the exceptional set.
One of the most important problems in computational algebra is the computation of real solutions of a polynomial system. Lasserre, Laurent, and Rostalski introduced a numerical method to compute the real radical of an ideal defining a finite set of real points. Their method is based on moment matrices and reduces the problem to solving a semidefinite program. In this work, we build on this method and on the work of Laurent and Mourrain to exploit the sparsity of the input polynomials. For this purpose, we extend the theory of moment matrices to the setting of affine toric varieties. We prove that, for sparse polynomial systems, this approach allows us to recover real points on affine toric varieties. Moreover, we show that a projection of the spectrahedron associated with the semidefinite program is, in fact, a polytope, and we propose a variation of the method to compute only one real solution of the system.
Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.
The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$.
We study the canonical form $Ω$ as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial $\operatorname{adj}_P$ - as an important invariant in this context. More precisely, for a polytope $P$ we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of $P$ is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form $Ω_0$ and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if $P$ has positive degree drop. Using it we can prove that zonotopes can be characterized as the $d$-polytopes that have maximal possible degree drop $d-1$. We obtain a decomposition formula for $Ω_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $Ω_s$ that can distinguish higher values of the degree drop.