How to stab a polytope

We study the set of linear subspaces of a fixed dimension intersecting a given polytope. To describe this set as a semialgebraic subset of a Grassmannian, we introduce a Schubert arrangement of the polytope, defined by the Chow forms of the polytope’s faces of complementary dimension. We show that the set of subspaces intersecting a specified family of faces is defined by fixing the sign of the Chow forms of their boundaries. We give inequalities defining the set of stabbing subspaces in terms of sign conditions on the Chow form

On semi-c-periodic functions of type I and applications

In this paper, we study the (newly introduced) class of functions called semi-c-periodic functions of type I in a Banach space. We first investigate their basic properties, including a convolution result. Using a suitable norm we prove that the set of such functions is a Banach space. We then study the existence and uniqueness of semi-c-periodic mild solutions for both autonomous and non-autonomous linear evolution equations. We achieve the existence results using the Banach fixed point theorem and the principle of reductio

Global solvability of the Laplace equation in weighted Sobolev spaces

We consider a non-local boundary value problem for the Laplace equation in an unbounded strip, studying the weak and strong solvability of the problem within the framework of the weighted Sobolev space W1,pν with a Muckenhoupt weight. Utilising tools from non-harmonic analysis, we prove that any weak solution belonging to W2,pν is also a strong solution and satisfies the corresponding boundary conditions. It is worth noting that such problems do not fall within the scope of the general theory of elliptic equations and therefore require a specialized approach

Chow-Lam recovery

We study the conditions under which a subvariety of the Grassmannian may be recovered from certain of its linear projections.  In the special case that our Grassmannian is projective space, this is equivalent to asking when a variety can be recovered from its Chow form; the answer is "always" by work of Chow in 1937. In the general Grassmannian setting, the analogous question is when a variety can be recovered from its Chow-Lam form. We give both necessary conditions for recovery and families of examples where, in contrast with the projective case, recovery is not possible

The positive orthogonal grassmannian

The Plücker positive region OGr+(k,2k) of the orthogonal Grassmannian  emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian OGr+(k,n) for general values of k, n. We determine the boundary structure of the quadric OGr+(1,n) in Pn-1+ and show that it is a positive geometry. We show that OGr+(k,2k+1) is isomorphic to OGr+(k+1, 2k+2) and connect its combinatorial structure to matchings on [2k+2]. Finally, we show that in the case n > 2k+1, the positroid cells of Gr+(k,n) do not induce a CW cell decomposition of OGr+(k,n)

On all-path convex, gated and Chebyshev sets in graphs

We present new characterizations for trees, block graphs, and geodetic graphs using all-path convex, gated and Chebyshev sets. Specifically, we prove that trees are exactly the graphs in which all-path convexity is a convex geometry. Block graphs are characterized as graphs in which all balls are all-path convex (equivalently, gated), and geodetic graphs are exactly those graphs where all balls (equivalently, closed neighborhoods) are Chebyshev. Additionally, we prove that almost all graphs have geodesically convex Chebyshev sets, provide a characterization of bipartite graphs with connected Chebyshev sets, and establish a criterion for graphs with trivial Chebyshev sets in the class of graph joins. Finally, we show that graphs of odd order with maximal number of edges under the Seidel switching operation always have trivial Chebyshev sets

Logarithmic discriminants of hyperplane arrangements

A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space

What is Positive Geometry?

This article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage of its development. We discuss the problems addressed in the volume and report on the progress. We also list some open challenges.

Kinematic varieties for massless particles

We study algebraic varieties that encode the kinematic data for n massless particles in d-dimensional spacetime subject to momentum conservation. Their coordinates are  spinor brackets, which we derive from the Clifford algebra  associated to the Lorentz group.  This was proposed for d=5 in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices

Binary geometries from pellytopes

Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell\u27s numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron