M.Sc Student M.Sc Thesis Yifrach Yuval Regular Random Sections of Convex Bodies and the Random Quotient of Subspace Theorem Department of Mathematics PROF. Emanuel Milman

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}.