Lift — mass-opt dichotomy

HPS (4.1), $s=3$ : $\beta_3=\inf I_3/D_3$ on $D_3=P\cap H^{-1}\cap L_2$ . Theorem 4.2: radial measures achieve the inf for $s\le 3$ . After Newton,

Q=\frac{I}{D},\qquad g(r,u)=\frac{r^3+u^3}{2\max(r,u)},\qquad D=\int r^2\,dm.

q2 compact (R=12, n=22 faces): $Q\ge\gamma=0.899526$ on $D$ -aspect $\le 12$ . Five singular faces are residual skips; copositive, $\min m^\top Mm>0.004$ .

Mass-stationarity on a finite support (one Lagrange multiplier) gives $V(r_i)=(Q/2)(r_i^2+D)$ at every used atom. With $\mathrm{inf\,supp}=1$ and $\mathrm{sup\,supp}=R$ ,

M_{-1}=Q+(Q-1)D,\qquad M_3=(Q-1)R^3+QDR.

Both positive $\Rightarrow R<Q/(1-Q)$ $\Rightarrow Q>R/(R+1)$ . At $R=12$ , $Q>12/13>\gamma$ .

Any finitely atomic $m$ : mass-opt on the same radii exists (simplex compact). Either the used aspect is $\le 12$ and $Q\ge\gamma$ , or it is $\ge 12$ and $Q>12/13$ . So every atomic measure has $Q\ge\gamma$ .

A compactly supported radial probability is a weak limit of finite spherical shells, with $Q$ continuous ( $g$ bounded on $[a,b]^2$ , $D\ge a^2$ ). A finite- $D$ measure is a $Q$ -limit of compact truncations ( $r^2$ is UI for that one measure). Hence $\beta_3\ge\gamma$ and $\beta_3^{-1}\le 1.11170<1.1118<1.1185$ .

Not the aspect- $\le 4$ number $1.1087$ . Not the withdrawn $1.1168$ .

Replay: ../run_all.sh.