A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

Rademacher, Luis

doi:10.1112/s0025579315000133

Cited by 7 publications

(4 citation statements)

References 16 publications

(22 reference statements)

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“…where the last passage is the content of formula (26) above. Then N(x) = ∇Φ * V (x)/(n + 1) by Proposition 6.1, and by (25),…”

Section: The Floating Body Of a Cone And Self-convolutionmentioning

confidence: 87%

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Isotropic constants and Mahler volumes

Klartag

2018

Advances in Mathematics

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This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.

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“…where the last passage is the content of formula (26) above. Then N(x) = ∇Φ * V (x)/(n + 1) by Proposition 6.1, and by (25),…”

Section: The Floating Body Of a Cone And Self-convolutionmentioning

confidence: 87%

“…This conjecture holds true in two dimensions. See also Rademacher [25] for supporting evidence. Theorem 1.1 admits the following:…”

Section: Introductionmentioning

confidence: 99%

Isotropic constants and Mahler volumes

Klartag

2018

Advances in Mathematics

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“…More specifically, while the two algorithms have close performances on the 100-Cube (ours surpassed H&R by 1.03×), the H&R algorithm was outperformed on the 100-Simplex (45× faster), 100-S-Cube (1.2× faster), 10-Birkhoff (8.8× faster), 10-Cross (19.4× faster), 50-P-Simplex (31.9× faster), e-coli (1359.9× faster), iAB-RBC-283, and iAT-PLT-636 (∼ 22× faster). The reason for the poor performance of H&R on the simplex is its worst possible isotropic constant 8 over all simplicial polytopes (Rademacher, 2016). HOPS was also unable to sample from iAB-RBC-283 where it was not able to round the polytope, due to its geometry.…”

Section: Discussionmentioning

confidence: 99%

Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo

Chalkis¹,

Fisikopoulos²,

Papachristou³

et al. 2021

Preprint

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We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted to a convex polytope. We prove that, starting from a warm start, it mixes in O(κd 2 2 log(1/ε)) steps for a well-rounded polytope, ignoring logarithmic factors where κ is the condition number of the negative log-density, d is the dimension, is an upper bound on the number of reflections, and ε is the accuracy parameter. We also developed an open source implementation of ReHMC and we performed an experimental study on various high-dimensional data-sets. Experiments suggest that ReHMC outperfroms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample.

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“…We refer e.g. to [1], [28], [29], [30], [24], [12], [6], [27]. No such results are valid for Petty's conjecture.…”

Section: Introductionmentioning

confidence: 99%

On the shape of a convex body with respect to its second projection body

Saroglou

2015

Advances in Applied Mathematics

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We prove results relative to the problem of finding sharp bounds for the affine invariant. Namely, we prove that if K is a 3-dimensional zonoid of volume 1, then its second projection body Π 2 K is contained in 8K, while if K is any symmetric 3-dimensional convex body of volume 1, then Π 2 K contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids-established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperimetric problem and the best known lower bound up to date for P (K) in 3 dimensions. As byproduct of our methods, we establish an almost optimal lower bound for high-dimensional bodies of revolution.

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A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

Cited by 7 publications

References 16 publications

Isotropic constants and Mahler volumes

Isotropic constants and Mahler volumes

Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo

On the shape of a convex body with respect to its second projection body

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