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(A statement of the inequality may be found here: Rioul - Information-theoretic proofs of entropy power inequalities, Proposition 9, (65c).) … I am interested in knowing if there is an alternate proof of this result that does not rely on entropy inequalities, and uses linear algebraic tools instead. …

Any pointers/related problems/inequalities are very much welcome. …

Let $B := A \Lambda^{1/2}$, so that $A\Lambda A^T = BB^T$. Using the Cauchy-Binet formula for the determinant of $BB^T$, we obtain $$|BB^T| = \sum_{1 \leq i_1 < \dots < i_k \leq n} |B_{i_1 i_2 \dots …

By scaling if necessary, we may assume without loss of generality that $a^2p + b^2\bar p = 1$ Substituting $u = a^2p$, we can rewrite the 1-dimensional inequality as \begin{align*} f(u) := u\log \frac …

I think I managed to prove the entire inequality analytically. The whole proof is a bit long to post here (about 7 pages) and involves ugly looking expressions. I'll outline the general strategy I use …

EDIT: After much work I was able to reduce the inequality to a single variable function which I need to show is non-positive. That function is (for $0\leq p\leq\frac{1}{2}$) $$\frac{p^2(\log(p))^2 - ( …

Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true? \begin{align*} \left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \ …

The conjecture is wrong! It wasn't as complicated as I thought it was. A simple counter example is over $\mathbb{Z}_6$. Consider $H(X) = 1$ and $H(y) = 1 +\epsilon$ where $\epsilon$ is very small. …

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the additio …