The main result of the Thesis is a lower bound for the maximal possible number of facets of a 0/1 polytope in ${\mathbb{R}}^n$ . By definition, a 0/1 polytope is the convex hull of a subset of the vertices of $[0,1]^n$ .

In general, if $P$ is a polytope in ${\mathbb{R}}^n$ , we write $f_{n-1}(P)$ for the number of its facets. Let $g(n):=\max\big\{ f_{n-1}(P_n): P_n\;\hbox{a 0/1 polytope in}\;{\mathbb{R}}^n\big\}$ . Fukuda and Ziegler asked what the behaviour of $g(n)$ is as $n\rightarrow\infty$ . The best known upper bound to date is

$$g(n)\leq 30(n-2)!$$

(for $n$ large enough), which is established by Fleiner, Kaibel and Rote. Regarding lower bounds, a major breakthrough was made by Bárány and Pór who proved that $g(n)\geq\left (\frac{cn}{\log n}\right )^{n/4}$ , where $c>0$ is an absolute constant. We show that the exponent $n/4$ can in fact be improved to $n/2$ :

There exists a constant $c>0$ such that

$$g(n)\geq \left (\frac{cn}{\log n}\right )^{n/2}.$$

The existence of 0/1 polytopes with many facets is established by a refinement of the probabilistic method developed by Bárány and Pór. We work with $\pm 1$ polytopes (i.e., polytopes whose vertices are sequences of signs). Let $X_1,\ldots ,X_n$ be independent and identically distributed $\pm 1$ random variables, defined on some probability space $(\varOmega,\mathcal{F},\mathbb{P})$ , with distribution

$$\mathbb{P}(X=1)=\mathbb{P}(X=-1)=\tfrac{1}{2}.$$

Set $\vec{X}=(X_1,\ldots ,X_n)$ and, for a fixed $N$ satisfying $n < N\leq 2^n$ , consider $N$ independent copies $\vec{X}_1,\ldots ,\vec{X}_N$ of $\vec{X}$ . This procedure defines the random 0/1 polytope $K_N={\rm conv}\{\vec{X}_1,\ldots ,\vec{X}_N\}$ . Under some restrictions on the range of values of $N$ , we obtain a lower bound for the expected number of facets ${\mathbb{E}}[f_{n-1}(K_N)]$ , for each fixed $N$ :

There exist two positive constants $a$ and $b$ such that: for all sufficiently large $n$ , and all $N$ satisfying $n^a\leq N\leq \exp (bn)$ , one has that

$${\mathbb{E}}[f_{n-1}(K_N)]\geq \left (\frac{\log N}{a\log n}\right )^{n/2}.$$

For the lower bound for $g(n)$ one only has to choose $N=\lfloor \exp (bn/\log n)\rfloor$ .

The second part of the Thesis is related to the strong form of Sylvester's classical problem about random points uniformly distributed in plane convex regions. We prove the following two facts: (1) If $K$ is a plane convex body with area $\vert K\vert=1$ and if $A_K$ denotes the distribution function of the area of a random triangle in $K$ , then $A_K(\alpha )\geq A_{\Delta }(\alpha )$ for all $\alpha >0$ , where $\Delta$ is a triangle. If $A_K=A_{\Delta }$ then $K$ is a triangle. (2) If $K$ is a symmetric plane convex body with area $\vert K\vert=1$ and if $B_K$ denotes the distribution function of the area of a random symmetric parallelogram in $K$ , then $B_K(\alpha )\geq B_P(\alpha )$ for all $\alpha >0$ , where $P$ is a parallelogram. If $B_K=B_P$ then $K$ is a parallelogram.