skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Pizza and 2-Structures
Let $$\mathcal{H}$$ be a Coxeter hyperplane arrangement in $$n$$-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group $$W$$. Furthermore assume that the arrangement is not of type $$A_1^n$$. Let $$K$$ be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group $$W$$ and let $$a$$ be a point such that $$K$$ contains the convex hull of the orbit of the point $$a$$ under the group $$W$$. In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers $$T$$ of $$\mathcal{H}$$ of the volumes of the intersections $$T\cap(K+a)$$ is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called $$2$$-structures introduced by Herb to study discrete series characters of real reduced groups.  more » « less
Award ID(s):
2247382
PAR ID:
10503759
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Discrete & Computational Geometry
Volume:
70
Issue:
4
ISSN:
0179-5376
Page Range / eLocation ID:
1221 to 1244
Subject(s) / Keyword(s):
Primary 52B45 20F55 51F15 Secondary 51M20 51M25 Coxeter arrangements 2-structures dissections Pizza theorem reflection groups intrinsic volumes pseudo-root systems Bolyai–Gerwien Theorem
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, International Mathematics Research Notices)
    Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 
    more » « less
  2. ; (, Model Theory)
    Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that T is power bounded. Let K be a model of T equipped with a T-convex valuation ring O and a T-derivation ∂ such that ∂ is monotone, i.e., weakly contractive with respect to the valuation induced by O. We show that the theory of monotone T-convex T-differential fields, i.e., the common theory of such K, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call T∂-henselianity. We establish an Ax-Kochen/Ershov theorem and further results for monotone T-convex T-differential fields that are T∂-henselian. 
    more » « less
  3. ; ; ; (, Mathematical Proceedings of the Cambridge Philosophical Society)
    Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by $${(e_{(k,j)}^*)_j}$$ , for $$k\in\N$$ , let $$Z=\ell^\infty(X_k:k\kin\N)$$ be their l ∞ -sum, and let $$T:Z\to Z$$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 
    more » « less
  4. ; ; (, Forum of Mathematics, Sigma)
    Abstract Let G be a finite group. Let $H, K$ be subgroups of G and $$H \backslash G / K$$ the double coset space. If Q is a probability on G which is constant on conjugacy classes ( $$Q(s^{-1} t s) = Q(t)$$ ), then the random walk driven by Q on G projects to a Markov chain on $$H \backslash G /K$$ . This allows analysis of the lumped chain using the representation theory of G . Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $$GL_n(q)$$ onto a Markov chain on $$S_n$$ via the Bruhat decomposition. The chain on $$S_n$$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains with G a compact group are discussed. 
    more » « less
  5. (, International Mathematics Research Notices)
    Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email