Symmetrization of Brace Algebras

We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra

Symmetrization of brace algebra

summary:Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)$ coincides with the natural symmetric brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)^{as}$, the direct sum of spaces of antisymmetric maps $V^{\otimes k}\to V$

Proof of the double bubble curvature conjecture

An area minimizing double bubble in $\mathbb R^n$ is given by two (not necessarily connected) regions which have two prescribed $n$-dimensional volumes whose combined boundary has least $(n\!-\!1)$-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for $n=2,3,4$, but is, for general volumes, unknown for $n\ge 5$. Here, for arbitrary $n$, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in $\mathbb R^3$, as well as some new component bounds in $\mathbb R^n$