2018/05/28 by Milman, Emanuel, Neeman, Joe · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · doi:10.48550/arxiv.1805.10961
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose ℝn into q cells of prescribed (positive) Gaussian measure when 2 ≤ q ≤ n+1, is to use a "simplicial cluster", obtained from the Voronoi cells of q equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). In particular, the case q=3 confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose ℝn (n ≥ 2) into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an (n-2)-dimensional plane at 120∘ angles (forming a tripod or "Y" shape in the plane). The case q=2 recovers the classical Gaussian isoperimetric inequality. To establish the Multi-Bubble conjecture, we show that in the above range of q, stable regular clusters must have flat interfaces, therefore consisting of convex polyhedral cells (with at most q-1 facets). In the Double-Bubble case q=3, it is possible to avoid establishing flatness of the interfaces by invoking a certain dichotomy on the structure of stable clusters, yielding a simplified argument.