A Geometric Lower Bound Theorem

Adiprasito, Karim; Nevo, Eran; Samper, Jose Alejandro

doi:10.1007/s00039-016-0363-x

Cited by 6 publications

(11 citation statements)

References 27 publications

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“…Let us try to remedy this by translating Theorem 3.1 to a statement about simplicial homology. The results of this section refine observations by Kalai [Kal87] from graph cycles to general homology cycles; see also [ANS16a] for a generalization of his ideas for geometrically restricted homology cycles. We use toric chordality instead.…”

Section: Relation To Resolution Chordalitysupporting

confidence: 78%

“…Remark 6.6. This corollary alone can also be shown using the Mayer-Vietoris resolution of thě Cech complex of local stress spaces as in [ANS16a].…”

Section: Relation To Resolution Chordalitymentioning

confidence: 90%

“…Toric chordality is way to study models of intersection theory in a rather precise geometric way without going to the toric variety. In particular, the present research also has some application to conjectures detailing the interplay between toric algebraic geometry and approximation theory [ANS16a].…”

Section: Introductionmentioning

confidence: 93%

“…Theorem 6.2 alone can also be proven rather elegantly using the stratification of the toric variety by torus orbits, which can be thought of as a diagram of spaces built over the intersection poset of P . Studying the associated resolution of the Ishida complex gives the desired fact easily (compare also [ANS16a]). Lemma 6.1 can be proven this way as well with just a little more work.…”

Section: Relation To Resolution Chordalitymentioning

confidence: 99%

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Toric chordality

Adiprasito

2017

Journal de Mathématiques Pures et Appliquées

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We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil-McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of graph chordality to higher skeleta of simplicial complexes with a coordinatization over characteristic 0, leading us to a far-reaching generalization of Kalai's work on applications of rigidity of frameworks to polytope theory. In contrast to "homological" chordality, the notion that is usually studied as a higher-dimensional analogue of graph chordality, we will show that toric chordality has several advantageous properties and applications.Most strikingly, we will see that toric chordality allows us to introduce a higher version of Dirac's propagation principle.Aside from the propagation theorem, we also study the interplay with the geometric properties of the simplicial chain complex of the underlying simplicial complex, culminating in a quantified version of the Stanley-Murai-Nevo generalized lower bound theorem.Finally, we apply our technique to give a simple proof of the generalized lower bound theorem in polytope theory and prove the balanced generalized lower bound conjecture of Klee and Novik.2010 Mathematics Subject Classification. Primary 14M25, 05C38 ; Secondary 32S50, 52C25, 13F55.

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Section: Relation To Resolution Chordalitysupporting

confidence: 78%

“…Remark 6.6. This corollary alone can also be shown using the Mayer-Vietoris resolution of thě Cech complex of local stress spaces as in [ANS16a].…”

Section: Relation To Resolution Chordalitymentioning

confidence: 90%

Section: Introductionmentioning

confidence: 93%

Section: Relation To Resolution Chordalitymentioning

confidence: 99%

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Toric chordality

Adiprasito

2017

Journal de Mathématiques Pures et Appliquées

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“…Hence we might use the f-vector of ∆ as a test statistic for log-concavity. The study of such tests seems related to the approximation theory of convex bodies developed by Adiprasito, Nevo and Samper [1]. What does their "higher chordality" mean for statistics?…”

Section: The Samworth Bodymentioning

confidence: 99%

Geometry of Log-Concave Density Estimation

Robeva

Sturmfels

Uhler

2018

Discrete Comput Geom

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Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on R d that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems.

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Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures

Novik,

Zheng

2024

Discrete Comput Geom

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A Geometric Lower Bound Theorem

Cited by 6 publications

References 27 publications

Toric chordality

Toric chordality

Geometry of Log-Concave Density Estimation

Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures

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