Are hyperbolicity cones projectionally exposed?

Lourenço, Roshchina, and Saunderson proved that every hyperbolicity cone — the convex cone attached to a hyperbolic homogeneous polynomial — is amenable, a facial regularity property stronger than facial exposedness. Their paper raises the question of how hyperbolicity cones fare with respect to the stronger property of projectional exposedness, under which every face of a cone is the image of the cone under some (not necessarily orthogonal) linear projection. Combining their theorem with a companion result gives that all hyperbolicity cones of dimension at most four are projectionally exposed, a fact the authors note has no obvious algebraic explanation; the question is open in dimension five and above. A 2025 follow-up by the same authors constructed the first amenable cone that is not projectionally exposed, in dimension five, showing that amenability cannot settle the question in higher dimensions. The open question is whether every hyperbolicity cone is projectionally exposed.

Background

A homogeneous polynomial $p$ in $n$ real variables is hyperbolic with respect to a direction $e$ with $p(e) \neq 0$ if for every $x \in \mathbb{R}^n$ the univariate polynomial $t \mapsto p(x + te)$ has only real roots. The hyperbolicity cone $\Lambda_+(p,e)$ is the closure of the connected component of $\{x : p(x) \neq 0\}$ containing $e$ ; a classical theorem of Gårding asserts it is a closed convex cone. Hyperbolicity cones generalize spectrahedral cones, which are linear sections $\{x : \sum_i A_i x_i \succeq 0\}$ of the cone of positive semidefinite matrices (here $\succeq 0$ means positive semidefinite): if $p(x) = \det(\sum_i A_i x_i)$ with $\sum_i A_i e_i$ positive definite, then $p$ is hyperbolic and $\Lambda_+(p,e)$ is the spectrahedral cone. The generalized Lax conjecture asserts that, conversely, every hyperbolicity cone is spectrahedral; it is confirmed in dimension three (Helton–Vinnikov; Lewis–Parrilo–Ramana) but unresolved in general, so geometric properties of spectrahedral cones cannot simply be transferred to hyperbolicity cones.

A face of a convex cone $K$ is a convex subset $F \subseteq K$ such that whenever $x, y \in K$ and a relative interior point of the segment $[x,y]$ lies in $F$ , the whole segment lies in $F$ . Projectional exposedness, a notion that arose in Borwein and Wolkowicz's work on facial reduction — the algorithmic process of restoring strict feasibility of a conic optimization problem by exposing a suitable face of the constraint cone — and was studied systematically by Sung and Tam, requires that for every face $F$ of $K$ there exists a linear projection (an idempotent linear map $P$ , not necessarily orthogonal) with $P(K) = F$ . Projectional exposedness implies amenability — for each face $F$ there is $\kappa > 0$ with $\mathrm{dist}(x,F) \leq \kappa\, \mathrm{dist}(x,K)$ for all $x \in \mathrm{span}\, F$ , a property controlling error bounds for conic feasibility problems (Lourenço) — which in turn implies that the cone is nice (every face of the dual cone is the dual of some face of the cone, a property used in facial-reduction algorithms), and niceness in turn implies that the cone is facially exposed. All three implications are strict in general. The largest classes of hyperbolicity cones known to be projectionally exposed are the polyhedral cones, the symmetric cones (the classical self-dual family containing the positive semidefinite and Lorentz cones), and, by Gouveia–Ito–Lourenço (2025), all homogeneous cones, those whose automorphism group acts transitively on the relative interior.

Lourenço–Roshchina–Saunderson proved that every hyperbolicity cone is amenable, that every face of a hyperbolicity cone is again a hyperbolicity cone, and that the intersection of two hyperbolicity cones is a hyperbolicity cone. Since their companion paper shows that amenable cones are projectionally exposed in dimension at most four, all hyperbolicity cones of dimension at most four are projectionally exposed — a fact the authors describe as lacking an obvious algebraic explanation. However, the same authors later constructed, in dimension five, two projectionally exposed cones whose intersection is not projectionally exposed, and the first example of an amenable cone that is not projectionally exposed. Thus amenability and projectional exposedness coincide only up to dimension four and are distinct from dimension five onward, so the implication through amenability cannot decide the question for hyperbolicity cones in dimension five and above. Notably, whether even all spectrahedral cones are projectionally exposed is itself open.

Problem Statement

Decide whether every hyperbolicity cone is projectionally exposed: for every homogeneous polynomial $p$ hyperbolic with respect to a direction $e$ , and every face $F$ of the hyperbolicity cone $\Lambda_+(p,e)$ , does there exist an idempotent linear map $P$ on $\mathbb{R}^n$ (not necessarily an orthogonal projection) such that $P(\Lambda_+(p,e)) = F$ ? The answer is known to be yes for cones of dimension at most four (the dimension of a cone being the dimension of its linear span), via amenability; the open cases are dimension five and above, where amenability no longer implies projectional exposedness, so the question must be decided directly — either by a proof for all hyperbolicity cones (which would in particular settle the still-open special case of spectrahedral cones), or by an explicit hyperbolic polynomial whose hyperbolicity cone has a face that is not the image of the cone under any linear projection.

References

Bruno F. Lourenço, Vera Roshchina, James Saunderson, "Hyperbolicity cones are amenable", Mathematical Programming 204(1–2):753–764 (2024; published online 2023), DOI 10.1007/s10107-023-01958-0, arXiv:2102.06359, https://arxiv.org/abs/2102.06359
Bruno F. Lourenço, Vera Roshchina, James Saunderson, "Amenable cones are particularly nice", SIAM Journal on Optimization 32(3):2347–2375 (2022), DOI 10.1137/20M138466X, arXiv:2011.07745, https://arxiv.org/abs/2011.07745
Bruno F. Lourenço, Vera Roshchina, James Saunderson, "Inner approximations of convex sets and intersections of projectionally exposed cones", arXiv:2501.12717 (2025), https://arxiv.org/abs/2501.12717
João Gouveia, Masaru Ito, Bruno F. Lourenço, "Faces of homogeneous cones and applications to homogeneous chordality", arXiv:2501.09581 (2025), https://arxiv.org/abs/2501.09581
Claus Scheiderer, "Smooth hyperbolicity cones are second-order cone representable", arXiv:2509.17121 (2025), https://arxiv.org/abs/2509.17121
Bruno F. Lourenço, "Amenable cones: error bounds without constraint qualifications", Mathematical Programming 186:1–48 (2021), DOI 10.1007/s10107-019-01439-3, arXiv:1712.06221, https://arxiv.org/abs/1712.06221
J. William Helton, Victor Vinnikov, "Linear matrix inequality representation of sets", Communications on Pure and Applied Mathematics 60(5):654–674 (2007), DOI 10.1002/cpa.20155, https://doi.org/10.1002/cpa.20155
Adrian S. Lewis, Pablo A. Parrilo, Motakuri V. Ramana, "The Lax conjecture is true", Proceedings of the American Mathematical Society 133(9):2495–2499 (2005), DOI 10.1090/S0002-9939-05-07752-X, https://doi.org/10.1090/S0002-9939-05-07752-X
Masaru Ito, Bruno F. Lourenço, "Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations", SIAM Journal on Applied Algebra and Geometry 7(1):236–263 (2023), DOI 10.1137/22M1513964, https://doi.org/10.1137/22M1513964
Jon M. Borwein, Henry Wolkowicz, "Regularizing the abstract convex program", Journal of Mathematical Analysis and Applications 83(2):495–530 (1981), DOI 10.1016/0022-247X(81)90138-4, https://doi.org/10.1016/0022-247X(81)90138-4
Chen-Han Sung, Bit-Shun Tam, "A study of projectionally exposed cones", Linear Algebra and its Applications 139:225–252 (1990), DOI 10.1016/0024-3795(90)90401-w, https://doi.org/10.1016/0024-3795(90)90401-w
Tim Netzer, Raman Sanyal, "Smooth hyperbolicity cones are spectrahedral shadows", Mathematical Programming 153(1):213–221 (2015), DOI 10.1007/s10107-014-0744-6, https://doi.org/10.1007/s10107-014-0744-6