Solution
= Solution
For convex $f$, the <subgradient inequality> gives
$$
f(x)-f(x_1,\ldots,z,\ldots,x_n)
\leq\partial_if(x)(x_i-z).
$$
Taking the positive supremum over $z\in[0,1]$ and summing squares shows
$$
V^+(x)\leq\sum_i(\partial_if(x))^2\leq1.
$$
The <modified logarithmic Sobolev inequality> from part a with $v=1$ therefore gives
$$
\mathbb P(Z-\mathbb EZ\geq t)\leq e^{-t^2/2}.
$$