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1. Pizza and 2-Structures

   https://doi.org/10.1007/s00454-023-00600-2  

   Ehrenborg, Richard; Morel, Sophie; Readdy, Margaret (December 2023, Discrete & Computational Geometry)

   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. 

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2. The best ways to slice a polytope

   https://doi.org/10.1090/mcom/4006  

   Brandenburg, Marie-Charlotte; De_Loera, Jesus; Meroni, Chiara (July 2024, American Mathematical Society)

   We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible combinatorial types of sections and craft algorithms that compute optimal sections of the polytope according to various combinatorial and metric criteria, including sections that maximize the number of -dimensional faces, maximize the volume, and maximize the integral of a polynomial. Our optimization algorithms run in polynomial time in fixed dimension, but the same problems show computational complexity hardness otherwise. Our tools can be extended to intersection with halfspaces and projections onto hyperplanes. Finally, we present several experiments illustrating our theorems and algorithms on famous polytopes. 

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3. Geometric Aspects of the Jacobian of a Hyperplane Arrangement

   https://doi.org/10.1093/imrn/rnaf172  

   DiPasquale, Michael; Sidman, Jessica; Traves, Will (July 2025, International Mathematics Research Notices)

   Abstract An embedding of the complete bipartite graph $$K_{3,3}$$ in $$\mathbb{P}^{2}$$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an important example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $$K_{3,3}$$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. We develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that the example above suggests. In characteristic zero we show that there is a one-to-one correspondence between weak perspective representations of a hyperplane arrangement and polynomials of minimal degree in certain saturations of the Jacobian ideal of the arrangement, providing a connection to algebra. In this setting we can use duality theorems to explain how rigidity theory is reflected in the graded Betti numbers of the module of logarithmic derivations of a line arrangement. 

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4. Counting Real Roots in Polynomial-Time via Diophantine Approximation

   https://doi.org/10.1007/s10208-022-09599-z  

   Rojas, J. Maurice (November 2022, Foundations of Computational Mathematics)

   Peter Burgisser (Ed.)

   Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH ). We also discuss a number- theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. 

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5. An analogue of polynomially integrable bodies in even-dimensional spaces

   https://doi.org/10.1016/j.jmaa.2023.127071  

   Agranovsky, M; Koldobsky, A; Ryabogin, D; Yaskin, V. (January 2024, Journal of mathematical analysis and applications)

   A bounded domain $$K \subset R^n$$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $$K$$ with a hyperplane $$\Pi$$ polynomially depends on the distance from $$\Pi$$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $$n$$ is even, and if $$n$$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $$n$$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property. 

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