Polar Duality Research Articles

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Polar duality and the reconstruction of quantum covariance matrices from partial data

We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work. We apply our constructions to Gaussian quantum states which leads to a non-trivial generalization of Pauli’s reconstruction problem and we state a simple tomographic characterization of such states.

Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function [Formula: see text] and a feasible set [Formula: see text]) and (ii) a dynamical system (a map [Formula: see text]). Its goal is to minimize f over the set [Formula: see text] of initial conditions that forever remain in [Formula: see text] under g. The focus of this paper is on the case where the mathematical program is a linear program and where the dynamical system is either a known linear map or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to [Formula: see text]. Our inner approximations are optimized with respect to the objective function f, and their semidefinite characterization—which has a semidefinite constraint of fixed size—is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations to [Formula: see text] from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case. Funding: O. Günlük was partially supported by the Office of Naval Research [Grant N00014-21-1-2575]. This work was partially funded by the Alfred P. Sloan Foundation, the Air Force Office of Scientific Research, Defense Advanced Research Projects Agency [Young Faculty Award], the National Science Foundation [Faculty Early Career Development Program Award], and Google [Faculty Award].

Train stop scheduling problem: An exact approach using valid inequalities and polar duality

Consider the problem of minimizing the number of train stops on a particular rail line. The objective is to assign passengers of each origin-destination pair to different trains in such a way that train capacity and passenger demand constraints are satisfied with minimal stoppages. The literature refers to this problem as the train stop scheduling problem. The problem has been extensively studied for decades, yet an exact solution approach has not been proposed. This paper proposes several valid inequalities to strengthen the mixed-integer programming formulation and solve the problem exactly in a reasonable amount of CPU time. The concept of polar duality has been utilized to find more complex valid inequalities which may be hard to find otherwise. Despite the problem’s high practical relevance, valid inequalities for the problem have not yet been studied in the literature. An aggregation procedure has been proposed to solve large size problem instances exactly. Computational study demonstrate the efficacy of the proposed valid inequalities.

Framed duality and mirror symmetry for toric complete intersections

This paper is devoted to systematically extend f-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in [17], to families of complete intersections subvarieties. Namely, f-mirror symmetry is induced by framed duality of framed toric varieties extending Batyrev-Borisov polar duality between Fano toric varieties. Framed duality has been defined and essentially well described for families of hypersurfaces in toric varieties in the previous [17]. Here it is developed for families of complete intersections, allowing us to strengthen some previous results on hypersurfaces. In particular, the class of projective complete intersections and their mirror partners are studied in detail. Moreover, a (generalized) Landau-Ginzburg/Complete-Intersection correspondence is discussed, extending to the complete intersection setup the LG/CY correspondence firstly studied Chiodo-Ruan and Krawitz.

Pointillisme à la Signac and Construction of a Quantum Fiber Bundle Over Convex Bodies

We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink.

Matching centroids by a projective transformation

Given two subsets of mathbb {R}^d, when does there exist a projective transformation that maps them to two sets with a common centroid? When is this transformation unique modulo affine transformations? We study these questions for 0- and d-dimensional sets, obtaining several existence and uniqueness results as well as examples of non-existence or non-uniqueness. If both sets have dimension 0, then the problem is related to the analytic center of a polytope and to polarity with respect to an algebraic set. If one set is a single point, and the other is a convex body, then it is equivalent by polar duality to the existence and uniqueness of the Santaló point. For a finite point set versus a ball, it generalizes the Möbius centering of edge-circumscribed convex polytopes and is related to the conformal barycenter of Douady-Earle. If both sets are d-dimensional, then we are led to define the Santaló point of a pair of convex bodies. We prove that the Santaló point of a pair exists and is unique, if one of the bodies is contained within the other and has Hilbert diameter less than a dimension-depending constant. The bound is sharp and is obtained by a box inside a cross-polytope.

Symplectic Polar Duality, Quantum Blobs, and Generalized Gaussians

We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson–Schrödinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Gaussian states in terms of partial information on the covariance ellipsoid, which allows us to formulate statements related to symplectic tomography.

Polar duality between pairs of transversal Lagrangian planes; applications to uncertainty principles

We extend the notion of polar duality to pairs (ℓ,ℓ′) of transversal Lagrangian planes in the standard symplectic space. (R2n,ω). This allows us to show that polar duality has a natural symplectic interpretation. Our main results are the following: we first show that the oblique projections Ωℓ and Ωℓ′ on ℓ and ℓ′ of a centrally symmetric convex body Ω containing a symplectic ball with radius one satisfy the duality relation Ωℓo⊂Ωℓ′. We thereafter show that if, conversely, (Ωℓ,Ωℓ′)⊂ℓ×ℓ′ is such that Ωℓo=Ωℓ′ then Ωℓ and Ωℓ′ are the projections of a symplectic ball with radius one which is the John ellipsoid of Ωℓ×Ωℓ′o. We apply these results to the quantum principle of indeterminacy, which contains as a particular case the usual uncertainty principle.

Remarks on Rigidity Properties of Conics

Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval, and the second concerns properties of the circle map associated with an oval and two pencils of lines.

An extension of polar duality of toric varieties and its consequences in Mirror Symmetry

The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called \emph{framed} duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.

The Pauli Problem for Gaussian Quantum States: Geometric Interpretation

We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way.

Directed fixed charge multicommodity network design: A cutting plane approach using polar duality

We present an efficient cutting-plane based approach to exactly solve a directed fixed charge network design (DFCND) problem, wherein the valid inequalities to the problem are generated using the polar duality approach. The biggest challenge in using this approach arises in constructing the polar dual of the problem. This would require enumerating all the extreme points of the convex hull of DFCND, which is computationally impractical for any instance of a reasonable size. Moreover, the resulting polar dual would be too large to solve efficiently, which is required at every iteration of the cutting-plane algorithm. The novelty of our solution approach lies in suggesting a way to circumvent this challenge by instead generating the violated facets, using polar duality, of the smaller substructures involving only a small subset of constraints and variables, obtained from 2-, 3-and 4-partitions of the underlying graph. For problem instances based on sparse graphs with zero flow costs, addition of these inequalities closes more than 20% of the optimality gap remaining after the addition of the knapsack cover inequalities used in the literature. This allows us to solve the problem instances in less than 400 s, on average, which otherwise take around 1000 s with the addition of only the knapsack cover inequalities, and around 4 hours for the Cplex MIP solver at its default setting.

Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $\Delta$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $\Delta$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $\Delta$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(\Delta, \Delta^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs $(\Delta, \Delta^{\vee})$ show that the stringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and $X^{\vee}$ may not satisfy the expected topological mirror symmetry test: $e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov's pairs $(\Delta, \Delta^\vee)$.

Families of Calabi–Yau hypersurfaces in [formula omitted]-Fano toric varieties

We provide a sufficient condition for a general hypersurface in a Q-Fano toric variety to be a Calabi–Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund–Hübsch–Krawitz construction in case the ambient is a Q-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi–Yau hypersurfaces which includes both Batyrev and Berglund–Hübsch–Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes Δ1⊆Δ2, where Δ1 and Δ2⁎ are canonical.

Two geometric interpretations of the multidimensional Hardy uncertainty principle

Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry. • We state a multidimensional version of Hardy's uncertainty principle. • The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid. • We express this result in terms of the projections of the Hardy ellipsoid on the x - and p -spaces.

Mirror Symmetry and Polar Duality of Polytopes

This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope.

THE VOLUME OF RANDOM POLYTOPES CIRCUMSCRIBED AROUND A CONVEX BODY

Let be a convex body in which slides freely in a ball. Let denote the intersection of closed half-spaces containing whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of and , and an asymptotic upper bound on the variance of the volume of . We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.

Quantum Indeterminacy and Polar Duality

The notion of polarity between sets, well-known from convex geometry, is a geometric version of the Fourier transform. We exploit this analogy to propose a new simple definition of quantum indeterminacy, using what we call “\(\hbar \)-polar quantum pairs”, which can be viewed as pairs of position-momentum indeterminacy with minimum spread. The existence of such pairs is consistent with the usual uncertainty principle, but is at the same time more general. We show that this quantum indeterminacy can be measured using the notion of symplectic capacity, which reduces to the notion of area in the phase plane.