Found problems: 5
Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality.
$$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$
Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.
Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i.
How many $\alpha$-cuts are there?
Let $C_1, ..., C_d$ be compact and connected sets in $R^d$, and suppose that each convex hull of $C_i$ contains the origin. Prove that for every i there is a $c_i \in C_i$ for which the origin is contained in the convex hull of the points $c_1, ..., c_d$.
In $R^d$ , all $n^d$ points of an n × n × ··· × n cube grid are contained in 2n - 3 hyperplanes. Prove that n ($n\geq3$) hyperplanes can be chosen from these so that they contain all points of the grid.
Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that
(a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and
(b) the Euclidean norm of every point $v$ in $A$ satisfies
\[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\]
Prove that the cardinality of $A$ is at most $2d+4$.