On the geometry of circumcentric directions of cones
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914515159875584 |
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| author | Bello-Cruz, Yunier |
| author_facet | Bello-Cruz, Yunier |
| contents | 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 $\norm{d}=\cosθ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26228 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the geometry of circumcentric directions of cones Bello-Cruz, Yunier Optimization and Control 90C25, 90C46, 52A20, 47H05, 90C22 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 $\norm{d}=\cosθ$ 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. |
| title | On the geometry of circumcentric directions of cones |
| topic | Optimization and Control 90C25, 90C46, 52A20, 47H05, 90C22 |
| url | https://arxiv.org/abs/2604.26228 |