  Particles interacting through long-range attraction and short-range repulsion
given by power-laws have been widely used to model physical and biological
systems, and to predict or explain many of the patterns they display. Apart
from rare values of the attractive and repulsive exponents $(\alpha,\beta)$,
the energy minimizing configurations of particles are not explicitly known,
although simulations and local stability considerations have led to conjectures
with strong evidence over a much wider region of parameters. For a segment
$\beta =2<\alpha<4$ on the mildly repulsive frontier we employ strict convexity
to conclude that the energy is uniquely minimized (up to translation) by a
spherical shell. In a companion work, we show that in the mildly repulsive
range $\alpha>\beta \ge2$, a unimodal threshold $2<\alpha_{\Delta^n}(\beta) \le
\max\{\beta,4\}$ exists such that equidistribution of particles over a unit
diameter regular $n$-simplex minimizes the energy if and only if $\alpha \ge
\alpha_{\Delta^n}(\beta)$ (and minimizes uniquely up to rigid motions if strict
inequality holds). At the point $(\alpha,\beta)=(2,4)$ separating these
regimes, we show the minimizers all lie on a sphere and are precisely
characterized by sharing all first and second moments with the spherical shell.
Although the minimizers need not be asymptotically stable, our approach
establishes $d_\alpha$-Lyapunov nonlinear stability of the associated
($d_2$-gradient) aggregation dynamics near the minimizer in both of these
adjacent regimes -- without reference to linearization. The
$L^\alpha$-Kantorovich-Rubinstein distance $d_\alpha$ which quantifies
stability is chosen to match the attraction exponent.
