Background from Convex Geometry

AbstractWe 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.

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Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Forum of Mathematics Sigma ◽

10.1017/fms.2021.10 ◽

2021 ◽

Vol 9 ◽

Author(s):

Joseph Malkoun ◽

Peter J. Olver

Keyword(s):

Convex Hull ◽

Convex Geometry ◽

Gauss Map ◽

Parameter Family ◽

Convex Hulls ◽

Dimensional Sphere ◽

Continuous Maps ◽

Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

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Translating between the representations of a ranked convex geometry

Discrete Mathematics ◽

10.1016/j.disc.2021.112399 ◽

2021 ◽

Vol 344 (7) ◽

pp. 112399

Author(s):

Oscar Defrain ◽

Lhouari Nourine ◽

Simon Vilmin

Keyword(s):

Convex Geometry

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Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽

10.1515/anly-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Siran Li

Keyword(s):

Banach Space ◽

Discrete Geometry ◽

Thin Shell ◽

Euclidean Geometry ◽

Convex Geometry ◽

Regularity Theory ◽

Elliptic Partial Differential Equations ◽

High Dimensional ◽

Space Theory ◽

Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

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Harmonic Analysis and Uniqueness Questions in Convex Geometry

Springer Proceedings in Mathematics & Statistics - Recent Advances in Harmonic Analysis and Applications ◽

10.1007/978-1-4614-4565-4_26 ◽

2012 ◽

pp. 327-337 ◽

Cited By ~ 3

Author(s):

Dmitry Ryabogin ◽

Vlad Yaskin ◽

Artem Zvavitch

Keyword(s):

Harmonic Analysis ◽

Convex Geometry

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Erratum to: Quotient metrics with applications in convex geometry

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry ◽

10.1007/s13366-012-0115-5 ◽

2012 ◽

Vol 53 (2) ◽

pp. 579-580

Author(s):

Agnieszka Bogdewicz ◽

Irmina Herburt ◽

Maria Moszyńska

Keyword(s):

Convex Geometry

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Decomposable Convexities in Graphs and Hypergraphs

ISRN Combinatorics ◽

10.1155/2013/453808 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Francesco M. Malvestuto

Keyword(s):

Convex Hull ◽

Convex Sets ◽

Nonempty Subset ◽

Convex Set ◽

Convex Geometry ◽

Convexity Space ◽

Vertex Set ◽

Convex Geometries ◽

Acyclic Hypergraph ◽

Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

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