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Let $f:\mathbb{P}(X) \rightarrow \mathbb{R}$ be a convex function where $\mathbb{P}(X)$ is the set of probability distribution on (finite) set $X$.

I am looking for condition on $f$ so I can said that the uniform disribution is minimizer of $f$.

For example, if there exist a group $G$ that act on $\mathbb{P}(X)$ such that for all $g \in G$: $f(p)=f(g\cdot p)$ then it follow that:

$$ f(\frac{1}{|G|}\sum_{g\in G} g\cdot p) \leq \frac{1}{|G|}\sum_{g\in G} f(g\cdot p) = f(p)$$ for all $p \in \mathbb{P}(X)$. If in addition $\frac{1}{|G|}\sum_{g\in G} g\cdot p$ is uniform on $X$ then the uniform disribution gives the minimum of $f$.

Is there a relaxaed condition to make sure that uniform is optimal?

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A affine map $g$ on $\mathbb P(X)$ corresponds to a (right) stochastic matrix $T$, i.e. a matrix with real nonnegative entries, each row summing to $1$, acting on $\mathbb P(X)$ (represented by row vectors) by $\pi \mapsto \pi T$. The matrix $T$ is doubly stochastic iff the uniform distribution is invariant under $g$. If in addition $T$ is aperiodic and irreducible, then the iterates $g^{(n)}(\pi)$ approach the uniform distribution as $n \to \infty$. Thus:

If $f(\pi T) \le f(\pi)$ for all $\pi \in \mathbb P(X)$, where $T$ is an aperiodic irreducible doubly stochastic matrix, then the uniform distribution minimizes $f$.

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  • $\begingroup$ nice! So I need some kind of mixing operation that reduce the function value. $\endgroup$ – nir Mar 24 '15 at 21:00

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