6 problems

Critical-volume conjecture. For $m\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\mathcal{E}(\cdot)$ among measurable sets $F\subset\mathbb{R}^n$ with $|F|=m$, and for…

Concentricity conjecture. Under the assumptions of the main theorem, there exists $\bar n>0$ such that for $n>\bar n$ minimizers $\Omega_n$ of $F_{\lambda_n,\sigma_n}$ are, up to t…

Ball-minimizer conjecture. The minimizer of $\widetilde E_\infty$ with fixed mass is given by a ball whenever such a minimizer exists.

Ball-minimizer conjecture. If a minimizer of $\widetilde E_\infty$ with fixed mass exists, then it is given by a ball.

Let $n=2$, let $\alpha\in(0,2)$, and define $f(m)=P(B[m])+\mathcal{V}_\alpha(B[m])$, where $B[m]$ is a ball of volume $m$. Set $m_0=0$, and for $k\geq1$ let $m_k$ be the critical m…

For $m>0$, let … where … A minimizer is a characteristic function representing a droplet of mass $m$. Critical-mass conjecture. There exists a critical mass $m_c$ such that minimiz…