Hi, I have some questions regarding this subject. The main reference I used is the paper "Displacement convexity of entropy and related inequalities on graphs" / nathael gozlan, cyril roberto, paul-marie samson, prasad tetali.
1) for a meaure $\mu$ on {0,1}, what is the formal definition of the n-fold product $\mu^{\otimes n}$?
Some background before continuing (need to take a big breath)
Theorem 6.4 in this paper states the discrete version of Perkopa Leindler:
Let n โ N$^*$, t โ [0, 1] and $\mu \in P(V^n)$. Suppose that ยต verifies the following property: for any $v_0, v_1 \in P(V^n)$, there exists a coupling $ \pi \in \Pi(v_0,v_1)$ such that
$$
H(v_t^\pi |\mu) \leq (1-t)H(v_0|\mu) + tH(v_1|\mu) - ct(1-t)I_2^{(n)}(\pi)
$$
If $( f , g, h)$ is a triple of functions on $V^n$ such that: $\forall x \in V^n , \forall m \in P(V^n),$
$$ (*)
\int \int h(z)v_t^{x,y}(dz) m(dy) \geq (1-t)f(x)+t \int g(y) m(dy)-c t(1-t)\sum_{i=1}^n (\int d(x_i,y_i)m(dy))^2
$$
then it holds
$$ (**)
\int e^{h(z)}\mu(dz) \geq (\int e^{f(x)}\mu(dx))^{1-t} (\int e^{g(y)}\mu(dy))^t
$$
and throrem 6.7 states the "Perkopa-Leindler for discrete hypercube": Let $\mu$ be a probability measure on {0,1}, $n \in N^{*}$. for all triple $(f,g,h)$ verifying (**) with c=1/2, it holds $$ \int e^{h(z)}\mu^{\otimes n}(dz) \geq (\int e^{f(x)} \mu^{\otimes n}(dx))^{1-t} (\int e^{g(y)} \mu^{\otimes n}(dy))^{t} $$
so now I can ask the following questions.
2) why do we need c = 1/2 in order to deduce theorem 6.7?
3) is (*) always true because of displacement convexity property?
