arXiv subjects

Raman Sanyal

Publications and source records attributed to Raman Sanyal.

At least 19 recordsLinked to original sources

arXiv2026

Indecomposability of 0/1-polytopes

We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes. As applications, we obtain uniform combinatorial indecomposability criteria for order and chain polytopes, matroid polytopes, stable set and clique polytopes, edge polytopes, flow polytopes, and 2-level/compressed polytopes. We also show that every nontrivial factorization of a multi-affine polynomial is a product of multi-affine polynomials in disjoint sets of variables.

math.CO↗

arXiv2024

Generalized angle vectors, geometric lattices, and flag-angles

Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincaré relation as the unique linear relation among interior angles. We show the existence and uniqueness of Euler--Poincaré-type relations for generalized angle vectors by building a bridge to the algebraic combinatorics of geometric lattices, generalizing work of Klivans--Swartz. We introduce flag-angles of polytopes as a geometric counterpart to flag-$f$-vectors. Flag-angles generalize the angle deficiencies of Descartes--Shephard, Grassmann angles, and spherical intrinsic volumes. Using the machinery of incidence algebras, we relate flag-angles of zonotopes to flag-$f$-vectors of graded posets. This allows us to determine the linear relations satisfied by interior/exterior flag-angle vectors.

math.CO↗

arXiv2024

Underlying Flag Polymatroids

We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying flag matroids to polymatroids. We show that the polytopes associated to underlying flag polymatroid are simple by proving that they are normally equivalent to certain nestohedra. We use this to show that polymatroids realized by subspace arrangements give rise to smooth toric varieties in flag varieties and we interpret our construction in terms of toric quotients. We give various examples that illustrate the rich combinatorial structure of flag polymatroids. Finally, we study general monotone paths on polymatroid polytopes, that relate to the enumeration of certain Young tableaux.

math.CO↗

arXiv2024

From linear programming to colliding particles

Although simplices are trivial from a linear optimization standpoint, the simplex algorithm can exhibit quite complex behavior. In this paper we study the behavior of max-slope pivot rules on (products of) simplices and describe the associated pivot rule polytopes. For simplices, the pivot rule polytopes are combinatorially isomorphic to associahedra. To prove this correspondence, we interpret max-slope pivot rules in terms of the combinatorics of colliding particles on a line. For prisms over simplices, we recover Stasheff's multiplihedra. For products of two simplices we get new realizations of constrainahedra, that capture the combinatorics of certain particle systems in the plane.

math.CO↗

arXiv2024

Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes

Kalai's $3^d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3^d$ faces. We give short proofs for two special cases: if $P$ is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if $P$ is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.

math.CO↗

arXiv2024

Inscribable fans I: Inscribed cones and virtual polytopes

We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $P$. We show that the associated space of polytopes, called the inscribed cone of $P$, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to $P$ is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension $2$ as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan $\mathcal{N}$ form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of $\mathcal{N}$. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be non-trivial even if $\mathcal{N}$ does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

math.MG↗

arXiv2022

Spectral Polyhedra

A "spectral convex set" is a collection of symmetric matrices whose range of eigenvalues form a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal-Sottile-Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity, and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies, and spectral zonotopes.

math.MG↗

arXiv2022

Inscribable Fans II: Inscribed zonotopes, simplicial arrangements, and reflection groups

An arrangement of hyperplanes is strongly inscribable if it has an inscribed (or ideal hyperbolic) zonotope. We characterize inscribed zonotopes and prove that the family of strongly inscribable arrangements is closed under restriction and localization. Moreover, we show that (strongly) inscribable arrangements are simplicial. We conjecture that only reflection arrangements and their restrictions are strongly inscribable and we verify our conjecture in rank-$3$ using the conjecturally complete list of irreducible simplicial rank-$3$ arrangements.

math.MG↗

arXiv2022

The Polyhedral Geometry of Pivot Rules and Monotone Paths

Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they are memory-less, in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. We show the existence of two polytopes, the pivot rule polytopes and the neighbotopes, that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences. We compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. Beyond optimization, our constructions provide new perspectives on classical geometric combinatorics. We introduce a normalized-weight pivot rule, we call the max-slope pivot rule which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine monotone path polytopes of Billera--Sturmfels. Moreover special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest improvement pivot rules we draw connections to sweep polytopes and polymatroids.

math.CO↗

arXiv2021

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to generalize the well-known Fisher--Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.

math.AG↗

arXiv2021

Fan Valuations and spherical intrinsic volumes

We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these 'intrinsic indicators' yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.

math.CO↗

arXiv2021

Coprime Ehrhart theory and counting free segments

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $α_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $α_2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.

math.CO↗

arXiv2021

Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums

In this paper we settle long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed faces of Minkowski sums. This has a wide range of applications and generalizes the classical the Upper Bound Theorems of McMullen and Stanley. Our main tool is relative Stanley--Reisner theory, a powerful generalization of the algebraic theory of simplicial complexes inaugurated by Hochster, Reisner, and Stanley. A key feature of our theory is the ability to accomodate topological as well as combinatorial restrictions. We illustrate this by providing several simplicial isoperimetric and reverse isoperimetric inequalities.

math.CO↗

arXiv2020

Geometric Inequalities for Anti-Blocking Bodies

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.

math.MG↗

arXiv2020

Unconditional reflexive polytopes

A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets

math.CO↗

arXiv2019

Tropical Carathéodory with Matroids

Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the tropical colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. In particular, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.

math.CO↗

arXiv2019

S-hypersimplices, pulling triangulations, and monotone paths

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.

math.CO↗

arXiv2019

On piecewise-linear homeomorphisms between distributive and anti-blocking polyhedra

Stanley (1986) introduced the order polytope and chain polytope of a partially ordered set and showed that they are related by a piecewise-linear homeomorphism. In this paper we view order and chain polytopes as instances of distributive and anti-blocking polytopes, respectively. Both these classes of polytopes are defined in terms of the componentwise partial order on $\mathbb R^n$. We generalize Stanley's PL-homeomorphism to a large class of distributive polyhedra using infinite walks in marked networks.

math.CO↗