The weighted Minkowski type existence theorem for C-pseudo-cones with given weighted surface area measure was proved by Schneider. In this paper, we prove the existence of a solution to the weighted L_p Minkowski problem for C-pseudo-cones in the case of p≥ 1 .
Hadwiger’s theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent k-ordering, for the existence of a hyperplane transversal for sets in ℝ^d . We obtain a quantitative generalization of Hadwiger’s theorem in ℝ^2 , showing that compact convex sets in ℝ^2 with a quantitative version of consistent ordering have a transversal satisfying quantitative requirements. Our proof generalizes the methods in Wenger’s proof of Hadwiger’s theorem in ℝ^2 . We also prove colorful versions of our results.
We study the geometric structure of Poncelet n-gons from a projective point of view. In particular we present explicit constructions of Poncelet n-gons for n = 7,8 and derive corresponding algebraic characterisations in terms of bracket polynomials, as well as describing a construction to produce a Poncelet 2n-gon from a starting Poncelet n-gon.
Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document} we find a connection between K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}-Lorentzian polynomials and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document} varies, even the set of quadratic K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}-Lorentzian and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}-completely log-concave polynomials coincide.
How do we cut a polygon into triangles that are all as “round” as possible, e.g., minimizing the maximum angle used? In this paper, we compute the optimal upper and lower angle bounds for triangulating an N-gon P with Steiner points, sharpening the 1960 theorem of Burago and Zalgaller that every polygon has an acute triangulation. For any polygon, we show both the upper and lower bounds can be computed in linear time from the list of interior angles of the polygon. We also show that both types of optimal bound are usually attained by some finite triangulation of the polygon (but sometimes they cannot both be attained by a single triangulation). We do not address the interesting problem of finding efficient triangulations that attain the optimal angle bounds; even in some simple cases, our construction gives many more triangles than are actually needed. The exceptional polygons where the optimal bounds can only be approximated, but not attained, are easily described: if and only if every interior angle is an integer multiple of 60^∘ , and some pair of sides has irrational length ratio. We also show that the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. This implies, in a stronger form, a 1984 conjecture of Gerver. Although the statements of our results involve only Euclidean geometry, the proofs depend on conformal and quasiconformal techniques.
Approximating convex bodies is a fundamental problem in geometry. Given a convex body K in ℝ^d for a fixed dimension d, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error ε . The best known uniform bound, due to Dudley (1974), shows that O(( diam (K)/ε )^(d-1)/2) facets suffice. Although this bound is optimal for fat objects, such as Euclidean balls, it is far from optimal for “skinny” convex bodies. Skinniness can be characterized relative to the Euclidean ball. Given a convex body K, define its area radius, arad (K) , to be the radius of the Euclidean ball having the same surface area as K. It follows from generalizations of the isoperimetric inequality that diam (K) ≥ 2 · arad (K) . We show that, given a convex body whose minimum width is at least ε , it is possible to approximate the body by a polytope having O(( arad (K)/ε )^(d-1)/2) facets. Our approach works by first reducing the problem of approximating convex bodies to that of approximating convex functions. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that there is a polar relationship between the Macbeath regions of a function and the Macbeath regions of its Legendre dual. This is combined with known bounds on the Mahler volume to bound the total size of the approximation.
A convex polytope P is monotypic if every polytope with the same set of outer normal vectors to its facets is isomorphic to P. Such a polytope P is also characterized by the fact that each non-empty intersection of P with a translate is homothetic to a summand of P. A strongly monotypic polytope P is such that the facet normals determine the isomorphism class of the arrangement of the hyperplanes spanned by its facets. It was shown in an earlier paper on the topic that, if P is strongly monotypic, then each non-empty intersection of P with a translate is a summand of P. However, the converse was left open. A proof of this has recently been given by Vuong Bui; however, a striking feature of the alternative proof given here, which as in the original paper uses representations, is that it reduces the problem to the already known 3-dimensional case.
Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of d-dimensional twinned chain polytopes is at most 6^d/2. In case d is even, the equality holds if and only if the polytope is isomorphic to a free sum of d/2 copies of del Pezzo polygons. This result contributes a partial answer to Nill's conjecture: the number of facets of a d-dimensional reflexive polytope is at most 6^d/2.
We study a problem of Santos about the largest possible diameter of a d-dimensional (abstract) simplicial complex on n vertices. For dimension 2, we determine the exact value of the maximum for every n using an explicit construction. We also come across a tantalizing open problem about the packing of squares of Hamilton cycles in the complete graph and obtain an infinite sequence of tight explicit constructions.
For integers 1< k < d-1 and r ⩾ k+2 , we establish new lower bounds on the maximum number of points in [n]^d such that no r lie in a k-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erdős and Purdy, improving Thiele’s bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for d ⩾ 4 . In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.
We generalize Pach and de Zeeuw’s lower bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains Ω (min{m^3/4n^3/4,m^2,n^2}) , as obtained from the proximity technique of Solymosi and Zahl.
Until recently, the simplest known flexible polyhedron embedded in Euclidean three-space was Steffen’s polyhedron on nine vertices. However, in 2024, an embedded flexible polyhedron on eight vertices was discovered. It attains the known lower bound for the number of vertices, showing that the simplest embedded flexible polyhedron has eight vertices. We introduce a method for making new flexible polyhedral surfaces from old ones. This general method applies to the above minimal example, giving another proof of its flexibility. We also construct a different, embedded flexible dodecahedron on eight vertices. This improves both the range of motion and the simplicity of the exposition.
We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in a fixed set of directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width constraint for a set of directions if the length of its orthogonal projection onto a line parallel to at least one of those directions is at most one. We analyze structural properties of the minimum number of pieces, focusing on monotonicity under polygon containment. Understanding this monotonicity is a crucial step toward designing efficient exact or approximation algorithms, as it dictates whether subproblems behave well under geometric decomposition. We show that the minimum partition number of a simple polygon is at least that of any subpolygon, provided that the subpolygon satisfies a certain orientation-wise convexity with respect to the polygon. As a core implication of this structural behavior, we prove a partition analogue of Bang’s conjecture for convex bodies in the plane: for any partition of a convex body in the plane, the sum of the relative widths of all parts is at least one. We also show that every convex polygon admits an optimal partition by parallel cuts. Moreover, given a convex polygon with n vertices in boundary order, an optimal partition into m pieces can be computed in O(mlog (1+n/m)) time.
If 𝒫 is a lattice polytope (i.e., 𝒫 is the convex hull of finitely many integer points in ℝ^d ), Ehrhart’s famous theorem (1962) asserts that the integer-point counting function |t 𝒫∩ℤ^d| is a polynomial in the integer variable t. Chapoton (2016) proved that, given a fixed integral form λ : ℤ^d →ℤ , there exists a polynomial cha_𝒫^λ (q,x) ∈ℚ(q)[x] such that the refined enumeration function ∑ _m∈ t 𝒫 q^λ (m) equals the evaluation cha_𝒫^λ (q, [t]_q) where, as usual, [t]_q:= q^t - 1 / q-1 ; naturally, for q=1 we recover the Ehrhart polynomial. Our motivating goal is to view Chapoton’s work through the lens of Brion’s Theorem (1988), which expresses the integer-point structure of a given polytope via that of its vertex cones. It turns out that this viewpoint naturally yields various refinements and extensions of Chapoton’s results, including explicit formulas for cha_𝒫^λ (q,x) , its leading coefficient, and its behavior as t →∞ . We also prove an analogue of Chapoton’s structural and reciprocity theorems for rational polytopes (i.e., with vertices in ℚ^d ).
Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone $\mathcal{K}$ we find a connection between $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone $\mathcal{K}$ varies, even the set of quadratic $\mathcal{K}$-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, $\mathcal{K}$-Lorentzian and $\mathcal{K}$-completely log-concave polynomials coincide.
We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, ℂP^2 , and the rational homology sphere QS^4(2) , where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations—which we deem interesting in their own rights.
The Reidemeister theorem states that any link in 3-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In this note we replace 3-space by any 3-manifold M and we extend the Reidemeister theorem (definition of the decoration and description of the combinatorial moves) in four situations, taking diagrams either of links or of bands (collections of cylinders and Möbius strips), either on an almost special spine or on a flow-spine of M. This partially reproves and extends a result of Brand, Burton, Dancso, He, Jackson and Licata.
