This page lists a few fun conjectures/problems and is updated from time to time. If you solve one, let me know!
Which Boolean Functions are Most Informative?
Conjecture: Let be i.i.d. Bernoulli(1/2), and let be the result of passing through a memoryless binary symmetric channel with crossover probability . For any Boolean function , we have
I am offering $100 for a correct proof of (or counterexample to) this conjecture. For more details on the conjecture itself, see my paper with Gowtham R. Kumar.
Entropy Power Inequality for a Triple Sum?
For a random vector with density on , define its entropy power as
If are independent random vectors, each having density on , is it true that
It turns out that this inequality is true if is Gaussian, but the only proof I know of is via a more general result (see Theorem 3 of this paper). So, I would also be interested in a direct proof of the inequality in the special case where is Gaussian.