M.Sc Thesis


M.Sc StudentYifrach Yuval
SubjectRegular Random Sections of Convex Bodies and the Random
Quotient of Subspace Theorem
DepartmentDepartment of Mathematics
Supervisor PROF. Emanuel Milman
Full Thesis textFull thesis text - English Version


Abstract

In this work, we extend several results in the Theory of Asymptotic Geometric Analysis regarding bounds on diameters of sections (and dually, in-radii of projections) of origin symmetric convex bodies.

For every origin symmetric convex body K in R^n and for every integer k\in [1,n/2], we prove that there exists a position, \overline K_k, of K, such that the geometric distances of (P_F\overline{K}_k)\cap E and P_E(\overline{K}_k\cap F) from the Euclidean ball in E are at most C\frac{n}{k}\log{\frac{n}{k}} with probability at least 1-2\exp(-ck), where F is randomised uniformly in G_{n,n-k} and E is a random subspace of F, of dimension n-2k (C,c>0 are universal constants).

We also prove that for every \alpha>1/2 there exists a position of K, denoted by \overline K_{\alpha} such that the measure of all subspaces F of R^n of co-dimension k-1 with \text{diam}(L\cap F)>\overline P_{\alpha}\left(\frac{n}{k}\right)^{\alpha} is at most \exp{-ck}, for all L=\overline{K}_{\alpha},\overline{K}^{\circ}_{\alpha} and k=1,\dots,n.

For an origin symmetric convex body, we denote M^*_k(K)=\int_{G_{n,k}}\text{diam}(P_F(K))d\sigma_{n,k}(F). We prove that for any 0<\eps,\delta<1/10, if n>\frac{c}{\eps^2} and:


\frac{M_{\floor*{cn\epsilon^2}}^*(K)}{a(K)}\frac{M_{\floor*{cn\epsilon^2}}^*(K^{\circ})}{a(K^{\circ})}>2-\delta/2,


then d_G(K,B_2^n)\leq 1(\epsilon\sqrt{\delta}) for some absolute constant c>0.

For an origin symmetric convex body K\subset \mathbb R^n and r>0, we define t(K,r) as the maximal 1\leq k\leq n such that the measure of all k-dimensional subspaces F with \text{diam}(K\cap F)>r is less than 3/4.

As a consequence of our Generalised Distance Lemma, we prove that:


t(K,r)(K^{\circ},\frac{1}{\kappa r})\geq c(1-\kappa)^2n


for every \kappa\in (0,1), origin symmetric convex body K\subset \mathbb R^n and r>0, whenever n>\frac{c}{(1-\kappa)^2}.