IMC 1997 · Problem 5

For a natural $n$ consider the hyperplane

$$R_0^n = \left\{ x = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n \;:\; \sum_{i=1}^{n} x_i = 0 \right\}$$

and the lattice $Z_0^n = \{ y \in R_0^n \;:\; \text{all } y_i \text{ are integers} \}$. Define the (quasi–)norm in $\mathbb{R}^n$ by $\|x\|_p = \left( \sum\limits_{i=1}^{n} |x_i|^p \right)^{1/p}$ if $0 < p < \infty$, and $\|x\|_{\infty} = \max\limits_i |x_i|$.

a) Let $x \in R_0^n$ be such that

$$\max_i x_i - \min_i x_i \le 1.$$

For every $p \in [1, \infty]$ and for every $y \in Z_0^n$ prove that

$$\|x\|_p \le \|x + y\|_p.$$

b) For every $p \in (0,1)$, show that there is an $n$ and an $x \in R_0^n$ with $\max\limits_i x_i - \min\limits_i x_i \le 1$ and an $y \in Z_0^n$ such that

$$\|x\|_p > \|x + y\|_p.$$