On the geometry of circumcentric directions of cones

Behling, Bello-Cruz, Lara-Urdaneta, Oviedo, and Santos showed that the circumcentric direction $d$ of a finitely generated polyhedral cone $\KK\subset\RR^n$ admits an inscribed Euclidean ball of radius $\norm{d}^2$ inside the polar cone $\Kpolar$. We sharpen this result in several ways. The exact set of admissible perturbations is a polyhedron, strictly larger than the inscribed ball off the generators and unbounded along $\Kpolar$. From it we read off a closed form for $\norm{d}^2$ in terms of the inverse Gram matrix of the conic base, with two-sided spectral bounds, and an aperture identity $\norm{d}=\cos\theta$ relating the generators to the axis $-d/\norm{d}$. The inscribed-ball estimate extends to closed convex pointed cones under one geometric condition: the normalized extremal section $E_\KK$ has affine hull avoiding the origin. The admissible set is then the intersection of half-spaces indexed by $E_\KK$, and the inscribed ball touches its boundary along $\norm{d}^2\,\closu E_\KK$. A Jordan-frame argument verifies the hypothesis for every simple symmetric cone and gives $\norm{d}^2=1/r$ for the Jordan rank $r$; the same value $1/n$ shows up for the doubly nonnegative cone, the direct-product case obeys the parallel-resistance rule $1/\norm{d}^2=\sum_\ell 1/\norm{d_\ell}^2$, and the $p$-cones with $p\ne 2$ provide a clean obstruction. We close with a sharp formula for the largest step from $d$ along a prescribed direction, worked out for $L_\infty$-ball constrained least squares and second-order cone programming; a piecewise smooth version where the inner Slater condition is exactly Mangasarian--Fromovitz; and a Bregman analogue covering a Mahalanobis instance and a mirror-descent step.