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 those people who are now not specialists within the box, could be capable of enjoy the handled subject matters. providing a presentation appropriate for pros with little historical past in research, geometry or chance, the paintings is going on to the relationship among isoperimetric-type inequalities and sensible inequalities, giving the reader quick entry to the center of those conjectures.
In addition, 4 contemporary and significant leads to this idea are provided in a compelling manner. the 1st are theorems because of Eldan-Klartag and Ball-Nguyen, referring to the variance and the KLS conjectures, respectively, to the hyperplane conjecture. subsequent, the most rules wanted end up 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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Additional resources for Approaching the Kannan-Lovász-Simonovits and Variance Conjectures
X/ dx, where denotes the convolution. x1 x2 /: we have hx1 x2 ; Âi2 D E hx1 ; Âi2 C E hx2 ; Âi2 D 2E hx; Âi2 : . 3 Let be a symmetric full dimensional isotropic log-concave probability on Rn . Then C p ; n where C > 0 is an absolute constant. The inequality is sharp and is attained when p is the uniform probability on nB2n . x/ dx, where V is a smooth, even and convex function. r/ D 0. r1 Â/ Ä r2 r1 r2 C r1 V . r2 Â/; V . r2 Â/g 2r2 2r2 whenever 0 < r1 < r2 . r2 Â/ if V is even or is the uniform probability in a convex set K, since in this case 0 2 K and r2 Â 2 K implies that r1 Â 2 K.
13 Let be a centered log-concave, probability on Rn . A/ In particular, this shows that constant. Ac /g: (iii) Let be the uniform probability on a convex body K. x/ the length of the maximum interval centered in x and contained in K. 11 It is clear how to deduce (ii) (and thus (i)) from E. Milman’s theorem. Indeed, let f 2 F , and x0 2 Rn . x0 /j x0 j: Thus, 1 2E jx x0 j Ä D1;1 Ä C p 1. x/ Ä diam K. Payne and Weinberger (see ) proved that if is the uniform probability on K p 1. / diam K for convex bodies with smooth boundary.
0. 6) • (iii) H) (ii) 1 Á2 D e2 . A/ 1=2. A/ 1=2 and let " > 0. k C 1/Á (if k D 0 the result is trivial). AkÁ /c . B kÁ / Ä 1=2. AkÁ /c / 1 1 1 2e2k 1 1 1 2ekC1 1 e 2 1 " Á p " p 1 16 1 e 2 and we obtain (ii). • (ii) H) (i) Assume that f W Rn ! f /. f /g. B/ 1=2. f / C tg 1 2ce dt : • (i) H) (iv) We can assume without loss of generality that kf kLips D 1. f /j D 2t fjf 0 Z 1 Ä 2a e 0 bt dt D 2a : b • (i) H) (iv) This implication will be proved in Sect. 5. 7 The following facts are true (i) In the case that D UK D aCK t > 0.