## Chapter 10 exercises

Let $\boldsymbol{X}$ be a random variable and let $1 \leq r \leq \infty$. Recall that the triangle (Minkowski) inequality implies that for real-valued functions $f_1, f_2$, $\|f_1(\boldsymbol{X}) + f_2(\boldsymbol{X})\|_r \leq \|f_1(\boldsymbol{X})\|_r + \|f_2(\boldsymbol{X})\|_r.$ More generally, if $w_1, \dots, w_m$ are nonnegative reals summing to $1$ and $f_1, \dots, f_m$ are real functions [...]

## §9.6: Highlight: The Kahn–Kalai–Linial Theorem

Recalling the social choice setting of Chapter 2.5, consider a $2$-candidate, $n$-voter election using a monotone voting rule $f : \{-1,1\}^n \to \{-1,1\}$. We assume the impartial culture assumption (that the votes are independent and uniformly random), but with a twist: one of the candidates, say $b \in \{-1,1\}$, is able to secretly bribe $k$ [...]