This paper addresses the so-called conformal capacities in $\mathbb R^n$,
$n\ge 3$, through comparing three existing definitions (due to Betsakos,
Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying
their associated iso-capacitary inequalities with connection to half-diameter,
mean-width, mean-curvature and ADM-mass, Hadamard type variational formula,
Minkowski type problem, and Yau type problem.Comment: 33 page