M. Fernandez V, G. V. Livshyts, S. Mui, On the smallest singular value of log-concave random matrices, preprint. See also slides.
A. V. Kolesnikov, G. V. Livshyts, L. Rotem, On p-Brunn-Minkowski and Brascamp-Lieb inequalities, preprint.
Julian Haddad, Dylan Langharst, Galyna V. Livshyts, Eli Putterman, On the polar of Schneider's difference body, submitted.
A. Colesanti, A. V. Kolesnikov, G. V. Livshyts, L. Rotem, On Weighted Blaschke-Santalo and strong Brascamp-Lieb inequalities, submitted.
A. Colesanti, E. Francini, G. V. Livshyts, P. Salani, The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction, to appear in Analysis and PDE.
O. Herscovici, G. V. Livshyts, L. Rotem, A. Volberg, Stability and the equality case in the B-theorem, to appear in Israel Journal of Math.
O. Herscovici, G. V. Livshyts, Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements, Proc. Amer. Math. Soc. 152 (2024), 4437-4450, DOI: https://doi.org/10.1090/proc/16889, Published electronically: August 23, 2024.
A.E. Litvak, G. V. Livshyts, New bounds on the minimal dispersion, J. of Complexity, 72 (2022), 101648.
G. Bonnet, D. Dadush, U. Grupel, S. Huberts, G. V. Livshyts, Combinatorial diameter of random polytopes, 38th Symposium on Computational Geometry (SoCG 2022).
G. V. Livshyts, A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures, Trans. Amer. Math. Soc. 376 (2023), 6663-6680.
Remark: an exciting improvement of the concavity power by a factor of n
was obtained by Giannopoulos, Eskenzis, Tziotziou in 2026. Moreover, the authors obtain a functional extension of this result. The authors communicated to me that they had difficulty with a subtle part of the argument from my paper ``A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures'', since in the proof of Proposition 4.2 yields the statement with $C_{poin}(A,\mu)$ on the right rather than $C_{poin}(K,\mu)$, as was stated. This is a minor technical point wchich was overlooked. Therefore, in the proof of the main result, the restriction of the measure on the set A should have been made isotropic, rather than the restriction on K itself. There is no problem with this, since A is just an intersection of a (convex) level set of a log-concave measure with a convex set, and the same straightforward argument applies to address this. I made minor corrections and uploaded the clarified version on arxiv.
G. V. Livshyts, On a conjectural symmetric version of Ehrhard's inequality, Trans. Amer. Math. Soc. 377 (2024), 5027-5085, DOI: https://doi.org/10.1090/tran/9177, Published electronically: May 21, 2024.
A. V. Kolesnikov, G. V. Livshyts, On the Local version of the Log-Brunn-Minkowski conjecture and some new related geometric inequalities,
International Mathematics Research Notices, 142, 08 June 2021. See also slides and
video
or this (better!) board and video.
Remark: Proposition 6.3 in the initial version was incorrect, see corrected version here:
updated version. This proposition is non essential and unrelated to any of the the main results.
J. Hosle, A. V. Kolesnikov, G. V. Livshyts, On the L_p-Brunn-Minkowski and dimensional Brunn-Minkowski conjectures for log-concave measures, The Journal of Geometric Analysis 31 (6), 5799-5836, (2020). See also slides and video.
J. Hao, H. Huang, G. V. Livshyts, K. Tikhomirov, Distribution of the minimal distance of random linear codes, IEEE Transactions on Information Theory 68 (10), 6388-6401, (2022).
G. V. Livshyts, K. Tikhomirov, R. Vershynin, The smallest singular value of inhomogeneous square random matrices, Ann. Probab. 49 (2021), no. 3, 1286-1309. See also slides.
G. V. Livshyts, Some remarks about the maximal perimeter of convex sets with respect to probability measures, Commun. Contemp. Math., (2020), https://doi.org/10.1142/S0219199720500376. See also slides.
B. Jaye, G. V. Livshyts, G. Paouris, P. Pivovarov, Remarks on the Renyi entropy of a sum of i.i.d. random variables, IEEE Trans. Inform. Theory 66 (2020), no. 5, 2898-2903. 94A17 (60E15).
G. V. Livshyts, The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding, Journal d'Analyse Mathematique, 145 (1), 257-306, (2021). See also slides (the same file as three rows above).
Bo'az Klartag, G. V. Livshyts, The lower bound for Koldobsky's slicing inequality via random rounding, Israel Seminar (GAFA) 2017-2019 Volume II, (2020), p. 43-63. See also some photos by Marton Naszodi.
[Erratum: in 2023, Julian Haddad spotted a technical error in the published version of the paper; this error has been fixed in the current arXiv version, posted July 18, 2023.]
A. V. Kolesnikov, G. V. Livshyts, On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type, Ad- vances in Mathematics 384 (2021) 107689. See also slides and video.
A. Colesanti, G. V. Livshyts, A note on the quantitative local version of the Log-Brunn-Minkowski inequality, Advances in Analysis and Geometry 2, special volume dedicated to the mathematical legacy of Victor Lomonosov, ISBN: 978-3-11-065339-7, (2020).
G. V. Livshyts, K. Tikhomirov, Cube is a strict local maximizer for the illumination number, Discrete and Computational Geometry, (2019), 63(1), 209-228.
G. V. Livshyts, An extension of Minkowski's theorem and its applications to questions about projections for measures, Advances in Mathematics, Volume 356, 7 November 2019, 106803. See also slides.
G. V. Livshyts, K. Tikhomirov, Randomized coverings of a convex body with its homothetic copies, and illumination, Proceedings of the AMS, (2019), ISSN 1088-6826(online), ISSN 0002-9939(print).
A. Colesanti, G. V. Livshyts, A. Marsiglietti, On the stability of Brunn-Minkowski type inequalities, Journal of Functional Analysis 273 (2017), no. 3, 1120-1139. See also video.
G. Livshyts, A. Marsiglietti, P. Nayar, A. Zvavitch, On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric type-inequalities, Transactions of the AMS, 369 (2017), no. 12, 8725-8742.
G. Livshyts, P. Paouris, P. Pivovarov, On sharp bounds for marginal densities of product measures, Israel Journal of Mathematics, (2016), 216(2), 877-889. See also slides.
G. V. Livshyts, Surface area of polytopes with respect to log-concave rotation invariant measures, Adv. Appl. Math., vol. 70, 54-69, (2015).
See also slides from my thesis defense.
G. V. Livshyts, Maximal Surface Area of a convex set in Rn with respect to log concave rotation invariant measures, GAFA Seminar Notes, 2116, (2014), 355-384. See also slides and video.
G. V. Livshyts, Maximal surface area of a convex set in Rn with respect to exponential rotation invariant measures, J. Math. Anal. Appl., 404, 2, (2013), 231-238.
(Image credit -- Pierre Bizeul)
Hi! I decided to create this tab in an attempt to battle the agonistic culture in Mathematics. In view of the recent developments, I believe that the concepts of attribution and credit for first discovery in Mathematics should be re-evaluated by the community. Many excellent papers appear on arXiv which combine the use of AI with brilliant human ideas, and this should continue. Many wonderful results involve human ideas only, and this should be championed as well. But what do we do with results obtained by mere prompting, with human input limited to what prompts to write, or which routes to explore, or deciding which results to pursue? I decided to merely post them here. I may revise their presentation and readability in the future. I do not intend to post this on arXiv, unless I combine this with additional human ideas and work. At the moment I do not have additional human ideas and feel a general sense of discoragement to pursue this. However, I find that it may be useful to share these insights with others. Shoot me an e-mail if you would like an Overleaf link (I am not posting overleaf links because the internet is big and my Overleaf is cozy but I will gladly share them upon request). Feel free to explore these, use it in your research, post your work. I am happy to hear thoughts, but you are not obliged to share them, and you are not obliged to reference these documents in case you pursue these directions. I will be extremely happy if others join this initiative of refraining from an arXiv race and sharing non-human math advances publicly without taking credit. I will be also very happy to hear philosophical input from others on this matter.
- Correlation-type Log-Brunn-Minkowski-type inequalities and other advances following the OpenAI proof of Log-Brunn-Minkowski
This document was generated from a series of prompts more or less amounting to: "please build theory around the new ideas of the proof of the Log-Brunn-Minkowski inequality". The specific explorations were not prompted but they use the memory across my chat history and the prior training I conducted over the course of the year on the relevant research explorations. I especially appreciate the progress which chat-gpt made on the Correlation-type inequality more general than the Log-Brunn-Minkowski inequality -- a beautiful question I learned back in the day from Dario Cordero-Erausquin.
- A simple proof of the B-conjecture
This is a very short and simple proof of the B-conjecture, a corollary of the OpenAI-solved Log-Brunn-Minkowski conjecture. It also involves additional results from the previous file, but the emphasis on outlining the short proof of the B-conjecture (which is shorter than Log-Brunn-Minkowski)
- About preferred directions
In this paper, Klartag discovered that an interesting property of eigenfunctions holds for measures with uniform lower bound on the hessian of the potential (which was an ingredient in his now-classical impoved Lichnerowitz inequality). Following the proof of Proposition 2.3 he called this "preferred directions" property and discussed its potential implications in case it was proved without the hessian lower bound. In this note, produced by chat-GPT following a series of prompts, it is shown that this property cannot hold without additional assumptions in the strongest possible sense. Additional explorations and a property for a packet of eigenfunctions equivalent to the now-proven KLS conjecture are discussed in the note.
- Unusual proofs of Ehhrard inequality
This note is a result of my attempts to produce novel proofs of Ehrhard's inequality with the help of chat-gpt. Latala showed that it suffices to prove a one-dimensional functional Ehrhard in order to deduce the full inequality, for it tensorizes. This note provides a PDE proof of this inequality, thereby giving a proof of the geometric Ehrhard itself (in conjunction with Latala's observations). This note also provides an equivalence of Ehrhard to an improved Gaussian Poincare inequality for linear functions via a cute dimensional lifting argument proposed by chat-gpt who drew his inspiration from Berwald, I think. This linear-functions Poincare is neat and can be also proved via PDE methods. I think in both cases these PDE methods draw inspirations from Borell. Still, I find this a curious story.
