New Maximally Stable Gaussian Partitions with Discrete Applications

Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures in hardness of approximation and social choice. Both conjectures generalize isoperimetric results by Borell on the heat kernel. One of the conjectures may be also be viewed as a generalization of the ”Double Bubble” theorem. The applications of the conjectures include an optimality result for majority in the context of Condorcet voting and a proof that the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal approximation factor assuming the Unique Games Conjecture. We finally derive a short proof of the first conjecture based on the extended Riesz inequality.