  The Seiberg-Witten solution to four-dimensional $\mathcal{N}=2$
super-Yang-Mills theory with gauge group $\text{SU}(N)$ and without
hypermultiplets is used to investigate the neighborhood of the maximal
Argyres-Douglas points of type $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$. A
convergent series expansion for the Seiberg-Witten periods near the
Argyres-Douglas points is obtained by analytic continuation of the series
expansion around the $\mathbb{Z}_{2N}$ symmetric point derived in
arXiv:2208.11502. Along with direct integration of the Picard-Fuchs equations
for the periods, the expansion is used to determine the location of the walls
of marginal stability for $\text{SU}(3)$. The intrinsic periods and K\"ahler
potential of the $(\mathfrak{a}_1,\mathfrak{a}_{N-1})$ superconformal fixed
point are computed by letting the strong coupling scale tend to infinity. We
conjecture that the resulting intrinsic K\"ahler potential is positive definite
and convex, with a unique minimum at the Argyres-Douglas point, provided only
intrinsic Coulomb branch operators with unitary scaling dimensions $\Delta>1$
acquire a vacuum expectation value, and provide both analytical and numerical
evidence in support of this conjecture. In all the low rank examples considered
here, it is found that turning on moduli dual to $\Delta \leq 1$ operators
spoils the positivity and convexity of the intrinsic K\"ahler potential.
