For regular anisotropies the lens cluster uniquely minimizes the anisotropic perimeter among (1,2)-clusters and the triod among (1,3)-clusters in R^2, with the result extended to general anisotropies by approximation.
Proof of the Double Bubble Conjecture
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We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \Bbb R^3.
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2026 1verdicts
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The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in $\mathbb{R}^2$
For regular anisotropies the lens cluster uniquely minimizes the anisotropic perimeter among (1,2)-clusters and the triod among (1,3)-clusters in R^2, with the result extended to general anisotropies by approximation.