By David Alonso-Gutiérrez, Jesús Bastero

Focusing on crucial conjectures of Asymptotic Geometric research, the Kannan-Lovász-Simonovits spectral hole conjecture and the variance conjecture, those Lecture Notes current the idea in an available means, in order that readers, even people who find themselves now not specialists within the box, may be capable of enjoy the handled subject matters. providing a presentation compatible for execs with little history in research, geometry or likelihood, the paintings is going on to the relationship among isoperimetric-type inequalities and useful inequalities, giving the reader fast entry to the center of those conjectures.

In addition, 4 contemporary and demanding ends up in this thought are awarded in a compelling method. the 1st are theorems because of Eldan-Klartag and Ball-Nguyen, touching on the variance and the KLS conjectures, respectively, to the hyperplane conjecture. subsequent, the most principles wanted turn out the simplest recognized estimate for the thin-shell width given through Guédon-Milman and an method of Eldan's paintings at the connection among the thin-shell width and the KLS conjecture are detailed.

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**Extra resources for Approaching the Kannan-Lovász-Simonovits and Variance Conjectures**

**Example text**

Var jxj2 / 4 Ä C E jxj. In the case of log-concave probabilities verifying the variance conjecture we obtain Var jxj2 Ä C1 2 E jxj2 Ä C n 4 : Then 1 1 Ä C2 n 4 Is. / and the corresponding Poincare’s inequality is 1 Var f Ä C n 2 2 E jrf j2 for any locally Lipschitz integrable function. 2 Concentration and Relation with Strong Paouris’s Estimate The next result shows the relation between Poincaré-type inequalities and thin-shell width and was established by Gromov and Milman. In particular, if KLS conjecture were true the mass in isotropic log-concave probabilities would be concentrated in a thin shell of constant width and depending with n as the central limit theorem suggests.

X/ a polynomial in n variables of degree d , p 1 and a log-concave probability (see [12]). n/ (recall that E f D E f ı T for any integrable function). n/. 7). n/ be any linear transformation. n/, where C is an absolute constant. n/ and D Œ 1 ; : : : ; n ( i > 0) is a diagonal map. n/ with the orthonormal basis fÁi gniD1 such that U1 U Ái D ei for all i . Á1 ; : : : ; Án / Z Z Z ::: D Sn 1 Sn 1 \Á? 1 Sn 1 \Á? \ 1 \Á? Á1 /; 54 1 The Conjectures is the Haar probability measure on S n 1 \ Á? \ Á?

1] or [50, Appendix III]). 9 Using E. Milman’s result we have an easy proof that the spectral gap for any log-concave probability in Rn is finite. Indeed, let f be any Lipschitz, integrable function in L2 . /. x/ Rn D 2krf k1 E jxj2 : Thus, p 1 Ä D2;1 Ä CD2;2 D C 0< p 1 . 6 Kannan-Lovász-Simonovits Spectral Gap Conjecture In this section we introduce the Kannan-Lovász-Simonovits spectral gap conjecture. We prove a certain type of linear invariance, which leads us to focus only on isotropic log-concave probabilities.