1 Introduction

A homogeneous polynomial p of degree d in n variables is hyperbolic with respect to \(e\in \mathbb {R}^n\) if \(p(e)>0\) and, for each \(x\in \mathbb {R}^n\) the univariate polynomial \(t\mapsto p(te-x)\) has only real zeros. Associated with a hyperbolic polynomial is the (closed) hyperbolicity cone

$$\begin{aligned} \Lambda _+(p,e) := \left\{ x\in \mathbb {R}^n\;:\;all zeros of t\mapsto p(te-x) are nonnegative \right\} . \end{aligned}$$
(1.1)

Hyperbolicity cones are convex cones that generalize spectrahedral cones. Spectrahedral cones are linearly isomorphic to the intersection of a cone of symmetric positive semidefinite matrices with a linear space. Indeed, if \(A_1,A_2,\ldots ,A_n\) are symmetric matrices and \(e\in {\mathbb {R}}^n\) is such that \(\sum _{i=1}^{n}A_ie_i\) is positive definite, then \(p(x) = \det (\sum _{i=1}^{n}A_ix_i)\) is hyperbolic with respect to e, and the associated hyperbolicity cone is the spectrahedral cone

$$\begin{aligned} \left\{ x\in {\mathbb {R}}^n:\;\sum _{i=1}^{n}A_ix_i \;\;\text {is positive semidefinite}\right\} . \end{aligned}$$

The generalized Lax conjecture asserts that every hyperbolicity cone is a spectrahedral cone. This is known to hold (in a stronger form) in dimension three [5, 6], but is unresolved in general. Examples of hyperbolic polynomials with hyperbolicity cones that are not currently known to be spectrahedral include mixed discriminants of partially specified tuples of symmetric matrices (see, e.g., [14, Section 6]), certain hyperbolic polynomials associated with (hyper)graphs [1], and certain cubic hyperbolic polynomials associated with the size of the largest clique in a graph [13].

If the generalized Lax conjecture were true, then any geometric property of spectrahedral cones should also hold for hyperbolicity cones. In this paper, we consider properties related to the faces (see Sect. 2.1 for a definition) of hyperbolicity cones that are known to hold for spectrahedral cones.

A closed convex cone \( {{\mathcal {K}}}\) is amenable if, for each face \( {\mathcal {F}}\) of \( {{\mathcal {K}}}\), there exists a constant \(\kappa >0\) such that

$$\begin{aligned} dist (x, {\mathcal {F}}) \le \kappa \, dist (x, {{\mathcal {K}}}) \quad for all x\in span ( {\mathcal {F}}) . \end{aligned}$$
(1.2)

The notion of amenable cones was introduced in [7] in the context of establishing error bounds for conic feasibility problems in the absence of constraint qualifications. In our context, we are interested in amenability because it is a strong notion of facial exposedness for convex cones.

Every amenable cone is nice (or facially dual complete) [7, Proposition 13], and every nice cone is facially exposed [9, Theorem 3]. All three notions coincide in dimension three (due to [10, Theorem 3.2]). However, in dimension four there are facially exposed cones that are not nice [12], and nice cones that are not amenable [8, Theorem 5.3]. Therefore amenability is a considerable strengthening of the property of being facially exposed.

It was recently established that spectrahedral cones are amenable [8, Corollary 3.5]. In light of the Lax conjecture, it is natural to ask whether hyperbolicity cones are also amenable. Our main result is the following.

Theorem 1.1

Every hyperbolicity cone is amenable.

Prior to this, it was known that hyperbolicity cones are facially exposed [11, Theorem 23], but it was not known whether every hyperbolicity cone is nice, let alone amenable. A simple consequence of Theorem 1.1 and [7, Proposition 13] is that every hyperbolicity cone is nice.

Another basic geometric fact about the faces of spectrahedral cones is that they are, themselves, spectrahedral cones. This follows from the fact that the faces of the positive semidefinite cone are positive semidefinite cones of smaller dimension. En route to the proof of Theorem 1.1, we show that faces of hyperbolicity cones are, themselves, hyperbolicity cones (Corollary 3.4). Based on this result, we also show that the intersection of two hyperbolicity cones is, again, a hyperbolicity cone (Corollary 3.5). This is a well-known fact under the additional assumption that the hyperbolicity cones share a relative interior point (and hence a direction of hyperbolicity). However, we could not find this result in the literature in the more subtle case where the relative interiors of the cones do not intersect.

Outline In Sect. 2.1, we recall the required definitions from convex geometry, and state a standard error bound result that will be needed for the proof of Theorem 1.1. In Sect. 2.2 we briefly summarize certain key facts and definitions about hyperbolic polynomials, hyperbolicity cones, and their derivative relaxations, that will be used throughout. In Sect. 3.1, we show that faces of hyperbolicity cones are, themselves, hyperbolicity cones, and that the intersection of two hyperbolicity cones is always a hyperbolicity cone. We establish these facts via a description of a face of a hyperbolicity cone as the (proper) intersection of the span of the face with an appropriate derivative relaxation of the hyperbolicity cone. We make this description completely explicit, in Sect. 3.2, by giving a concrete description of the span of a face of a hyperbolicity cone given a point in its relative interior. In Sect. 4 we use our representation of faces, together with the error bound stated in Sect. 2.1, to show that every hyperbolicity cone is amenable. In Sect. 5 we discuss some further questions related to the faces of hyperbolicity cones.

2 Preliminaries

2.1 Convex analysis and error bounds

Let \(C \subseteq {\mathbb {R}}^n\) be a closed convex set. In what follows we denote the interior, relative interior and the span of C by \({{\,\textrm{int}\,}}C\), \({{\,\textrm{ri}\,}}C\) and \({{\,\textrm{span}\,}}C\), respectively. We also recall that a face of C is a closed convex set \( {\mathcal {F}}\) contained in C satisfying the following property: whenever \(x,y \in C\) are such that \( \alpha x + (1-\alpha ) y \in {\mathcal {F}}\) for some \(\alpha \in (0,1)\), we have \(x,y \in {\mathcal {F}}\). In this case, we write \( {\mathcal {F}}\mathrel {\unlhd }C\).

In what follows, we assume that \({\mathbb {R}}^n\) is equipped with some norm \(\Vert {\cdot }\Vert \) and denote by \({{\,\textrm{dist}\,}}(x,C)\) the distance from x to C so that \({{\,\textrm{dist}\,}}(x,C) {:}{=}\inf \{\Vert {x-y}\Vert \mid y \in C \}.\) Finally, we need the following error bound result.

Proposition 2.1

(Linear regularity under a constraint qualification) Let L be a linear subspace and \( {{\mathcal {K}}}\) a closed convex cone with \(L\cap {{\,\textrm{int}\,}} {{\mathcal {K}}}\ne \emptyset \). Then there exists \(\kappa >0\) such that

$$\begin{aligned} {{\,\textrm{dist}\,}}(x,L\cap {{\mathcal {K}}}) \le \kappa ({{\,\textrm{dist}\,}}(x,L)+{{\,\textrm{dist}\,}}(x, {{\mathcal {K}}}))\quad \forall x\in {\mathbb {R}}^n. \end{aligned}$$

Proof

It follows from [2, Corollary 3] that L and \( {{\mathcal {K}}}\) are boundedly linearly regular, i.e. for every bounded set S there exists a \(\kappa _S>0\) that satisfies the error bound

$$\begin{aligned} {{\,\textrm{dist}\,}}(x,L \cap {{\mathcal {K}}})\le \kappa _S \max \{ {{\,\textrm{dist}\,}}(x,L), {{\,\textrm{dist}\,}}(x, {{\mathcal {K}}})\}\quad \forall x\in S. \end{aligned}$$
(2.1)

Moreover, because L and \( {{\mathcal {K}}}\) are closed convex cones, by homogeneity one can choose \(\kappa _S: = {\bar{\kappa }}\) in (2.1) independent of S (see [2, Theorem 10]). \(\square \)

2.2 Hyperbolic polynomials and hyperbolicity cones

In this section, we recall some basic facts about hyperbolic polynomials, hyperbolicity cones, and their derivative relaxations. Recall that if p is hyperbolic with respect to e then we write \(\Lambda _+(p,e)\) for the associated hyperbolicity cone. It turns out that the dependence of the hyperbolicity cone on the choice of e is quite weak. In fact, if \(z\in int (\Lambda _+(p,e))\) then p is hyperbolic with respect to z, and \(\Lambda _+(p,e) = \Lambda _+(p,z)\), a result of Gårding [4]. When p and e are clear from the context, we will omit them and write \(\Lambda _+\).

Polynomial inequality description of hyperbolicity cones The definition of the closed hyperbolicity cone \(\Lambda _+(p,e)\) given in (1.1) is expressed in terms of the nonnegativity of the zeros of \(p(te-x)\). It is also useful to work with an alternative description of \(\Lambda _+(p,e)\) in terms of polynomial inequalities. Let \(D_e^kp(x)= \left. \frac{d^k}{dt^k}p(x+te)\right| _{t=0}\) denote the kth directional derivative of p in the direction e. Expanding \(p(x+te)\) in powers of t gives

$$\begin{aligned} p(x+te) = p(x) + tD_ep(x) + \frac{t^2}{2{!}}D_e^2p(x) + \cdots + \frac{t^d}{d{!}}D_e^dp(x). \end{aligned}$$

Since a real-rooted polynomial has all roots non-positive if and only if it has all coefficients non-negative, we have

$$\begin{aligned} \Lambda _+(p,e) = \{x\in {\mathbb {R}}^n:\; p(x) \ge 0,\; D_ep(x) \ge 0,\; \ldots ,\; D_e^{d-1}p(x) \ge 0\}. \end{aligned}$$
(2.2)

(This description can also be obtained, for instance, by combining Proposition 18 and Theorem 20 of [11].) The interior of \(\Lambda _+(p,e)\) is then

$$\begin{aligned} int (\Lambda _+) = \{x\in {\mathbb {R}}^n:\; p(x)> 0,\; D_ep(x)> 0,\; \ldots ,\; D_e^{d-1}p(x) > 0\}. \end{aligned}$$
(2.3)

Derivative relaxations If p is hyperbolic with respect to e, then so are the directional derivatives \(D_e^mp(x)\) (by Rolle’s theorem, see also [11, Proposition 18]). Denote the associated hyperbolicity cone by \(\Lambda _+(D_e^mp,e)\). If p and e are clear from the context we write \(\Lambda _+^{(m)}{:}{=}\Lambda _+(D_e^mp,e)\). If \(m\le d-1\) then it follows from (2.2) that

$$\begin{aligned} \Lambda _+^{(m)} {=} \{x\in {\mathbb {R}}^n:\; D_e^mp(x) {\ge } 0,\; D_e^{m+1}p(x) {\ge } 0, \;\ldots ,\; D_e^{d-1}p(x) {\ge } 0\}. \end{aligned}$$
(2.4)

For convenience we define \(\Lambda _+^{(d)} = {\mathbb {R}}^n\). From this description it is clear that

$$\begin{aligned} \Lambda _+ \subseteq \Lambda _+^{(1)} \subseteq \cdots \subseteq \Lambda _+^{(d-1)} \subseteq \Lambda _+^{(d)} = {\mathbb {R}}^n. \end{aligned}$$
(2.5)

These are known as Renegar derivatives or, in light of (2.5), derivative relaxations of \(\Lambda _+\).

The following (standard) technical fact about the coefficients of univariate polynomials with nonnegative coefficients will be useful later in our arguments.

Lemma 2.2

Let \(p(t) = a_dt^d + a_{d-1}t^{d-1} + \cdots + a_1t + a_0\) be a univariate polynomial with real zeros, nonnegative coefficients, and \(a_d>0\). If \(a_k=0\) for some \(k<d\), then \(a_\ell = 0\) for all \(\ell < k\).

Proof

Because all coefficients are nonnegative and \(a _d > 0\), we have \(p(\lambda ) > 0\) for every positive \(\lambda \). Therefore, all roots of p are nonpositive. Let \(\lambda _1,\ldots ,\lambda _d\le 0\) be the roots of p.

We recall that \(|a_k/a_d| = |e_{d-k}(\lambda _1,\ldots ,\lambda _d)|\) is the absolute value of the elementary symmetric polynomial of degree \(d-k\) in the roots of p. Since all the roots of p are non-positive, \(|e_{d-k}(\lambda _1,\ldots ,\lambda _d)|= e_{d-k}(|\lambda _1|,\ldots ,|\lambda _d|)\). Therefore, if \(a_k = 0\), it follows that every product of \(d-k\) roots of p must vanish. This shows that \(a_{\ell } = 0\) for \(\ell < k\), because \(|a_{\ell }/a_d| = e_{d-\ell }(|\lambda _1|,\ldots ,|\lambda _d|)\) is the absolute value of a sum of products of more than \(d-k\) roots of p. \(\square \)

3 Faces of hyperbolicity cones

In this section, we show that the faces of hyperbolicity cones can be expressed as the intersection of a linear space with a suitable derivative relaxation of the hyperbolicity cone, in such a way that the linear space meets the interior of the derivative relaxation. This new representation of the faces of hyperbolicity cones is the crucial ingredient in our proof that every hyperbolicity cone is amenable, which is the focus of Sect. 4.

3.1 Facial structure of hyperbolicity cones

Given \(x\in {\mathbb {R}}^n\), let \(mult _{p}(x)\) denote the multiplicity of 0 as a zero of \(t\mapsto p(te-x)\). It turns out that this quantity is independent of the choice of e [11, Proposition 22], so we suppress e from the notation.

The following result tells us that a point of multiplicity m in a hyperbolicity cone is in the interior of the mth derivative relaxation.

Lemma 3.1

Let p be hyperbolic with respect to e. If \(mult _{p}(x)=m\) and \(x\in \Lambda _+(p,e)\) then

  • \(D_e^{k}p(x) = 0\) for \(k=0,1,\ldots ,m-1\) and

  • \(D_e^kp(x) > 0\) for \(k=m,m+1,\ldots , d\),

and so \(x\in int (\Lambda _+^{(m)})\).

Proof

Since \(mult _{p}(x) = m\), we know that \(t\mapsto p(te-x)\) vanishes to exactly order m at \(t=0\). Since \(p(te-x)= (-1)^dp(x+(-t)e)\), it also holds that \(t\mapsto p(x+te)\) vanishes to exactly order m at \(t=0\). Hence \(D_e^\ell p(x) = 0\) for \(\ell =0,1,\ldots ,m-1\), and and \(D_e^mp(x)\ne 0\) (otherwise \(mult _{p}(x) > m\)). As such,

$$\begin{aligned} p(x+te) = \frac{t^m}{m{!}}D_e^mp(x) + \frac{t^{m+1}}{(m+1){!}}D_e^{m+1}p(x) + \cdots + \frac{t^d}{d{!}}D_e^dp(x). \end{aligned}$$

Since \(x\in \Lambda _+\), we know that \(D_e^kp(x) \ge 0\) for all k and so, since \(D_e^mp(x)\ne 0\), it must be the case that \(D_e^mp(x) > 0\). By Lemma 2.2, it follows that \(D_e^kp(x) > 0\) for \(k=m,m+1,\ldots , d-1\). Moreover \(D_e^dp(x) = d{!}p(e) > 0\), since p is hyperbolic with respect to e. Since \(D_e^kp(x)>0\) for \(k=m,m+1,\ldots ,d-1\), it follows from (2.4) and (2.3) that \(x\in int (\Lambda _+^{(m)})\). \(\square \)

We also require the following key fact, which is part of [11, Theorem 26], about multiplicity and faces of hyperbolicity cones.

Lemma 3.2

Suppose that p is hyperbolic with respect to e. If \( {\mathcal {F}}\) is a face of \(\Lambda _+(p,e)\) and \(x,z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\) then \(mult _p(x) = mult _p(z)\).

We now establish the main result of this subsection, which gives a new description of the faces of hyperbolicity cones.

Proposition 3.3

(Faces and derivative relaxations) Suppose that p is hyperbolic with respect to e and \( {\mathcal {F}}\) is a face of \(\Lambda _+(p,e)\). Let \(z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\) and let \(m = mult _p(z)\). Then \(z\in int (\Lambda _+^{(m)})\) and

$$\begin{aligned} {\mathcal {F}}= {{\,\textrm{span}\,}}( {\mathcal {F}}) \cap \Lambda _+^{(m)}. \end{aligned}$$

Proof

Since \(mult _p(z) = m\) and \(z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\subseteq \Lambda _+\), it follows from Lemma 3.1 that \(p(z) = D_ep(z) = \cdots = D_e^{m-1}p(z) = 0\) and that \(z\in int (\Lambda _+^{(m)})\).

Since \(z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\) has multiplicity m, it follows from Lemma 3.2 that every point in the relative interior of \( {\mathcal {F}}\) has multiplicity m. As such \(p(x) = D_ep(x) = \cdots = D^{m-1}_ep(x) = 0\) for all \(x\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\). Since \({{\,\textrm{ri}\,}}( {\mathcal {F}})\) is full-dimensional in \({{\,\textrm{span}\,}}( {\mathcal {F}})\), it follows that \(p(x) = D_ep(x) = \cdots = D^{m-1}_ep(x) = 0\) for all \(x\in {{\,\textrm{span}\,}}( {\mathcal {F}})\).

Since \( {\mathcal {F}}\) is a face of \(\Lambda _+\), we have that \( {\mathcal {F}}= {{\,\textrm{span}\,}}( {\mathcal {F}}) \cap \Lambda _+\). It follows from (2.5) that \( {\mathcal {F}}= {{\,\textrm{span}\,}}( {\mathcal {F}}) \cap \Lambda _+ \subseteq {{\,\textrm{span}\,}}( {\mathcal {F}})\cap \Lambda _+^{(m)}\). For the reverse inclusion, suppose that \(x\in {{\,\textrm{span}\,}}( {\mathcal {F}})\cap \Lambda _+^{(m)}\). Then \(p(x) = D_e^1p(x) = \cdots = D_e^{m-1}p(x) = 0\) (since \(x\in {{\,\textrm{span}\,}}( {\mathcal {F}})\)) and \(D_e^kp(x)\ge 0\) for \(k=m,m+1,\dots ,d-1\) (since \(x\in \Lambda _+^{(m)}\)). Hence \(x\in \Lambda _+\) (from (2.2)) and so \({{\,\textrm{span}\,}}( {\mathcal {F}})\cap \Lambda _+^{(m)} \subseteq {{\,\textrm{span}\,}}( {\mathcal {F}})\cap \Lambda _+ = {\mathcal {F}}\). \(\square \)

This result can be rephrased as saying that faces of hyperbolicity cones are, themselves, hyperbolicity cones. To see this, first we clarify the notation we will use. If \(L \subseteq \mathbb {R}^n\) is a subspace and p is hyperbolic with respect to \(e\in L\), then the restriction of p to L, denoted \(\left. p\right| _{L}\), is also hyperbolic with respect to e when thought of as a polynomial on L. The corresponding hyperbolicity cone, \(\Lambda _+(\left. p\right| _L,e)\), is linearly isomorphic to \(\Lambda _+(p,e) \cap L\).

Corollary 3.4

Suppose that p is hyperbolic with respect to e and \( {\mathcal {F}}\) is a face of \(\Lambda _+(p,e)\). If \(z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\) and \(m=mult _p(z)\) then \(q = \left. D_e^{m}p\right| _{{{\,\textrm{span}\,}}( {\mathcal {F}})}\) is hyperbolic with respect to z and \( {\mathcal {F}}= \Lambda _+(q,z)\).

We can use Corollary 3.4 to understand the intersection of two hyperbolicity cones. Let \(\Lambda _+(p_1,e_1)\) and \(\Lambda _+(p_2,e_2)\) be two hyperbolicity cones. If their relative interiors intersect, then \(\Lambda _+(p_1,e_1)\cap \Lambda _+(p_2,e_2)\) is also a hyperbolicity cone and the corresponding hyperbolic polynomial is \(p_1p_2\) restricted to the span of \(\Lambda _+(p_1,e_1)\cap \Lambda _+(p_2,e_2)\). However, when \(({{\,\textrm{ri}\,}}\Lambda _+(p_1,e_1))\cap ({{\,\textrm{ri}\,}}\Lambda _+(p_2,e_2)) = \emptyset \), the situation is less obvious. As a consequence of Corollary 3.4, we can show that the intersection of two hyperbolicity cones is a hyperbolicity cone.

Corollary 3.5

Suppose that \(p_1\) and \(p_2\) are hyperbolic polynomials with respect to \(e_1\) and \(e_2\), respectively. Then, \(\Lambda _+(p_1,e_1)\cap \Lambda _+(p_2,e_2)\) is a hyperbolicity cone.

Proof

Let \( {{\mathcal {K}}}= \Lambda _+(p_1,e_1)\cap \Lambda _+(p_2,e_2)\). It is a well-known fact from convex analysis that if \( {\mathcal {F}}_i\) is the smallest face of \(\Lambda _+(p_i,e_i)\) containing \( {{\mathcal {K}}}\), then we have

$$\begin{aligned} {{\,\textrm{ri}\,}} {{\mathcal {K}}}\subseteq {{\,\textrm{ri}\,}} {\mathcal {F}}_i \qquad for i =1,2, \end{aligned}$$

and \( {{\mathcal {K}}}= {\mathcal {F}}_1 \cap {\mathcal {F}}_2\). (See, for example, [8, Proposition 2.2].) In particular, there exists \(z \in {{\,\textrm{ri}\,}} {{\mathcal {K}}}\) such that \(z \in ({{\,\textrm{ri}\,}} {\mathcal {F}}_1)\cap ({{\,\textrm{ri}\,}} {\mathcal {F}}_2)\).

For \(i = 1,2\), let \(m_i {:}{=}mult _{p_i}(z)\) and \(q_i {:}{=}\left. D_{e_i}^{m_i}p_i\right| _{{{\,\textrm{span}\,}}( {\mathcal {F}}_i)}\). By Corollary 3.4, \(q_i\) is hyperbolic with respect to z and \( {\mathcal {F}}_i = \Lambda _+(q_i,z)\). If \(L = {{\,\textrm{span}\,}} {{\mathcal {K}}}\), then \(\left. (q_1q_2)\right| _{L}\) is hyperbolic with respect to z and the corresponding hyperbolicity cone is \( {{\mathcal {K}}}= {\mathcal {F}}_1\cap {\mathcal {F}}_2\). \(\square \)

3.2 Explicit description of the span of a face

In Sect. 3.1 we gave a description of a face of a hyperbolicity cone as the intersection of the span of the face with an appropriate derivative relaxation. While Proposition 3.3 will be sufficient to establish that hyperbolicity cones are amenable, the expression therein is implicit in the sense that in describing a face, it refers to the span of that face. In this section we show how to give an explicit description of the span of a face of a hyperbolicity cone by using results from [11]. The description only uses the hyperbolic polynomial, a direction of hyperbolicity, and a point in the relative interior of the face.

Suppose that q is hyperbolic with respect to e and \(z\in \Lambda _+(q,e)\) has \(mult _q(z) = 1\). This means that z is a smooth point of the boundary of the hyperbolicity cone. Following [11] let

$$\begin{aligned} T_z\Lambda _+(q,e) = \{x:\; D_xq(z) = 0\} \end{aligned}$$
(3.1)

denote the tangent space to the hyperbolicity cone at the point z.

Our explicit description of the span of a face is closely related to the following result of Renegar.

Theorem 3.6

(Renegar [11, Proposition 9 and Theorem 10]) Suppose that \(z\in \Lambda _+(q,e)\) and \(mult _q(z)=1\) and \(x\in T_z\Lambda _+(q,e)\). Then \(D^2_xq(z) \le 0\). Moreover, exactly one of the following holds:

  • \(q(z+tx) = 0\) for all t and there exists \(\varepsilon >0\) such that \(z+tx \in \Lambda _+(q,e)\) and \(mult _q(z+tx)=1\) whenever \(t\in (-\varepsilon ,\varepsilon )\);

  • \(D_x^2q(z) < 0\).

At this point, it is convenient to introduce one more piece of notation. Suppose that \(z\in \Lambda _+(q,e)\) has multiplicity one with respect to q and that the associated tangent space is \(T_z\Lambda _+(q,e)\). We use the notation \(\left. H_q(z)\right| _{T_z}:T_z\Lambda _+(q,e)\rightarrow T_z\Lambda _+(q,e)\) to denote the self-adjoint linear map associated with the restriction of the quadratic form \(x\mapsto D^2_xq(z)\) to the subspace \(T_z\Lambda _+(q,e)\). (See Sect. 3.3 for concrete examples.) Equivalently, we can also think of \(\left. H_q(z)\right| _{T_z}\) as the Hessian of the polynomial \({\tilde{q}} = \left. q\right| _{T_z\Lambda _+(q,e)}\) obtained by restricting q to the subspace \(T_z\Lambda _+(q,e)\).

We now give an explicit description of the span of a face of a hyperbolicity cone. This can be combined with Proposition 3.3 to give an explicit description of a face of a hyperbolicity cone, given a point in the relative interior of the face.

Proposition 3.7

Suppose that p is hyperbolic with respect to e and \( {\mathcal {F}}\) is a face of \(\Lambda _+(p,e)\). Let \(z\in {{\,\textrm{ri}\,}}( {\mathcal {F}})\), let \(m= mult _p(z)\), and let \(q(x) = D^{m-1}_ep(x)\). Then

$$\begin{aligned} {{\,\textrm{span}\,}}( {\mathcal {F}}) = \{x\in T_z\Lambda _+(q,e) :\; \left. H_q(z)\right| _{T_z} x = 0\}. \end{aligned}$$
(3.2)

Proof

If \( {\mathcal {F}}\) is a face of \(\Lambda _+(p,e)\) of multiplicity m then \( {\mathcal {F}}\) is also a face of \(\Lambda _+(q,e)\) of multiplicity 1 [11, Proposition 25]. As such, it is enough to establish the result when \(m=1\) and \(p=q\).

For the “\(\supseteq \)” inclusion, assume that \(x\in T_z\Lambda _+(q,e)\) and that \(\left. H_q(z)\right| _{T_z}x = 0\). It follows that \(D_x^2q(z) = 0\). Since \(z\in \Lambda _+(q,e)\) and \(mult _q(z)=1\), it follows from Theorem 3.6 that \(q(z+tx) = 0\) for all t and that there exists \(\varepsilon >0\) such that \(z+tx\in \Lambda _+(q,e)\) for all \(t\in (-\varepsilon ,\varepsilon )\). Let \({\hat{\varepsilon }} = \varepsilon /2\). Since \( {\mathcal {F}}\) is a face of \(\Lambda _+(q,e)\), \(z\pm {\hat{\varepsilon }}x \in \Lambda _+(q,e)\) and \(2z = (z-{\hat{\varepsilon }} x)+ (z+{\hat{\varepsilon }} x)\), it follows that \(z-{\hat{\varepsilon }} x,z+{\hat{\varepsilon }} x\in {\mathcal {F}}\). Taking a linear combination of these points we see that \([(z+{\hat{\varepsilon }} x)-(z-{\hat{\varepsilon }} x)]/(2{\hat{\varepsilon }}) = x\in {{\,\textrm{span}\,}}( {\mathcal {F}})\).

We now consider the reverse inclusion. First, we will check that

$$\begin{aligned} {\mathcal {F}}\subseteq T_z\Lambda _+(q,e). \end{aligned}$$
(3.3)

Let \(y \in {\mathcal {F}}\). Since \(z+ty\in {\mathcal {F}}\) for all \(t\ge 0\) and \( {\mathcal {F}}\) is a face of multiplicity one, it follows that \(z+ty\) has multiplicity at least one for all \(t \ge 0\). This implies that \(q(z+ty) = 0\) for all \(t \ge 0\). Since q is a polynomial, we have that, in fact, \(q(z+ty) = 0\) for all \(t\in \mathbb {R}\). Therefore \(D_yq(z) = 0\), which shows that \(y \in T_z\Lambda _+(q,e)\), by the definition of \(T_z\Lambda _+\) in (3.1). Furthermore, we can also conclude that \(D_y^2q(z) = 0\).

It remains to show that \(y\in {\mathcal {F}}\) implies that \(\left. H_q(z)\right| _{T_z}y = 0\). Now, the quadratic form \(x\mapsto D_x^2q(z)\) is negative semidefinite restricted to \(T_z\Lambda _+(q,e)\) (from Theorem 3.6) and \(y \in T_z\Lambda _+(q,e)\) by (3.3). Therefore, \(D_y^2q(z) = 0\) implies that y is in the nullspace of the associated linear map, which is \(\left. H_q(z)\right| _{T_z}\). This establishes that \( {\mathcal {F}}\) is contained in the right-hand side of (3.2). Since the right-hand side of (3.2) is a subspace, it follows that it also contains \({{\,\textrm{span}\,}} {\mathcal {F}}\).\(\square \)

3.3 Illustrative examples

Fig. 1
figure 1

On the left is an affine slice of \(\Lambda _+(q,e)\), showing the points \({\tilde{z}}\) and \({\tilde{w}}\) in the relative interiors of the faces of interest. On the right are (rotations of) the same affine slices of \(\Lambda _+(q,e)\), the relaxation \(\Lambda _+(D_eq,e)\), and the spans of the two faces of interest

Full size image

In this section we apply the results of Propositions 3.3 and 3.7 to give explicit descriptions of two faces of the hyperbolicity cone associated with the polynomial

$$\begin{aligned} q(x_1,x_2,x_3,x_4) = (x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4)(x_1+x_2+x_3-x_4/2). \end{aligned}$$

The polynomial q is hyperbolic with respect to \(e=(1/4,1/4,1/4,1/4)\). For the purposes of visualization, we consider the three dimensional affine section \(x_1+x_2+x_3+x_4=1\) of the hyperbolicity cone \(\Lambda _+(q,e)\), under the affine isomorphism that sends the standard basis vectors in \({\mathbb {R}}^4\) to the vertices of a regular tetrahedron. The resulting convex body is shown in Fig. 1. It is the intersection of the elliptope with an affine halfspace.

As a first example, consider the face with \({\tilde{z}}\) in its relative interior, shown in Fig. 1. This corresponds to a three-dimensional face of the cone \(\Lambda _+(q,e)\) with the point \(z = (1/9,1/9,1/9,2/3)\) in its relative interior. This point has multiplicity one with respect to q, because only the linear factor of q vanishes at z. Proposition 3.3 tells us that the face is the intersection of its span with the derivative relaxation \(\Lambda _+(D_eq,e)\). This derivative relaxation, together with the span of the face, are shown in Fig. 1.

It is clear from Fig. 1 that the span of the face is just the hyperplane corresponding to the linear factor in q, i.e., \(\{(x_1,x_2,x_3,x_4):\ x_1+x_2+x_3-x_4/2=0\}\). We can also obtain this from Proposition 3.7. Since

it follows that \(T_{z}\Lambda _+(q,e) = \{(x_1,x_2,x_3,x_4): x_1+x_2+x_3-x_4/2 = 0\}\). Consequently the restriction of \(x\mapsto D_x^2q(z)\) to \(T_z\Lambda _+(q,e)\) is identically zero, and so \(\left. H_q(z)\right| _{T_z} = 0\). Hence Proposition 3.7 tells us that the span of the face is just \(T_z\Lambda _+(q,e)\) in this case.

As a second example, consider the face with \({\tilde{w}}\) in its relative interior, shown in Fig. 1. This corresponds to a two-dimensional face of \(\Lambda _+(q,e)\) with the point \(w = (1/2,0,0,1/2)\) in its relative interior. This point has multiplicity one with respect to q since the cubic factor of q vanishes to order one at w and the linear factor does not vanish at w. Again Proposition 3.3 tells us that the face is the intersection of its span with the derivative relaxation \(\Lambda _+(D_eq,e)\), as shown in Fig. 1.

We now show how to compute the span of this face using Proposition 3.7. Since

$$\begin{aligned} \nabla q(w) = \frac{1}{16} \begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix} \quad and \quad D_x^2q(w) = \frac{3x_1}{4}\left( x_2+x_3\right) + \frac{1}{2}\left( x_2^2+3x_2x_3+x_3^2\right) \end{aligned}$$

it folllows that \(T_w\Lambda _+(q,e)= \{(x_1,x_2,x_3,x_4): x_2+x_3=0\}\). If we express elements of \(T_w\Lambda _+(q,e)\) in the form \((y_1,y_2,-y_2,y_3)\) for \(y\in {\mathbb {R}}^{3}\) then we have that

$$\begin{aligned} D_{(y_1,y_2,-y_2,y_3)}^2q(w) = -y_2^2/2\quad and so \quad \left. H_w(q)\right| _{T_w} = \begin{bmatrix} 0 &{} 0 &{} 0\\ 0 &{} -1/2 &{} 0\\ 0 &{} 0 &{} 0\end{bmatrix}. \end{aligned}$$

Clearly, then, \(\left. H_w(q)\right| _{T_w}y = 0\) if and only if \(y_2=0\). From Proposition 3.7 we get that

$$\begin{aligned} span ({\mathcal {F}}){} & {} = \{(y_1,y_2,-y_2,y_3): y_2=0,\; y_1\in {\mathbb {R}},\;y_3\in {\mathbb {R}}\} \\{} & {} =\{(y_1,0,0,y_3): y_1\in {\mathbb {R}},\;y_3\in {\mathbb {R}}\} \end{aligned}$$

which is just the span of the first and last coordinate directions.

4 Amenability of hyperbolicity cones

In this brief section we establish that every hyperbolicity cone is amenable (Theorem 1.1). The argument combines the standard error bound from Proposition 2.1 and our new representation of faces of hyperbolicity cones from Proposition 3.3.

Proof of Theorem 1.1

Let p be hyperbolic with respect to \(e\in {\mathbb {R}}^n\), and let \( {\mathcal {F}}\mathrel {\unlhd }\Lambda _+(p,e)\) be a face of the associated hyperbolicity cone. By Proposition 3.3, there exists some derivative relaxation \(\Lambda _+^{(m)}\) satisfying the following two properties:

$$\begin{aligned} {\mathcal {F}}\cap ({{\,\textrm{int}\,}}\Lambda _+^{(m)} ) \ne \emptyset \quad and \quad {\mathcal {F}}= {{\,\textrm{span}\,}}( {\mathcal {F}}) \cap \Lambda _+^{(m)}. \end{aligned}$$

In particular, \({{\,\textrm{span}\,}}( {\mathcal {F}}) \cap ({{\,\textrm{int}\,}}\Lambda _+^{(m)} ) \ne \emptyset \), and so by Proposition 2.1, there exists \(\kappa > 0\) such that for all \(x\in \mathbb {R}^n\) we have

$$\begin{aligned} {{\,\textrm{dist}\,}}(x, {\mathcal {F}})&\le \kappa \max ({{\,\textrm{dist}\,}}(x,{{\,\textrm{span}\,}} {\mathcal {F}}),{{\,\textrm{dist}\,}}(x, \Lambda _+^{(m)} ))\\&\le \kappa \max ({{\,\textrm{dist}\,}}(x,{{\,\textrm{span}\,}} {\mathcal {F}}),{{\,\textrm{dist}\,}}(x, \Lambda _+(p,e)), \end{aligned}$$

where the last inequality follows from the fact that \(\Lambda _+(p,e) \subseteq \Lambda _+^{(m)}\). This implies that

$$\begin{aligned} {{\,\textrm{dist}\,}}(x, {\mathcal {F}}) \le \kappa {{\,\textrm{dist}\,}}(x, \Lambda _+(p,e)) \quad for all x \in {{\,\textrm{span}\,}} {\mathcal {F}}. \end{aligned}$$

Therefore \(\Lambda _+(p,e)\) is amenable. \(\square \)

5 Discussion

It could be fruitful to examine in more depth the geometry and facial structure of hyperbolicity cones and check how it compares with spectrahedral cones. One recent result in this style is that the tangent cone to a hyperbolicity cone at a point is, again, a hyperbolicity cone [14, Theorem 5.9], which parallels the analogous result for spectrahedral cones. For a direction where many questions remain open, it might be interesting to see how spectrahedral cones and hyperbolicity cones fare with regards to projectional exposedness.

We say that a closed convex cone \( {{\mathcal {K}}}\) is projectionally exposed if for every face \( {\mathcal {F}}\mathrel {\unlhd } {{\mathcal {K}}}\) there exists a projection P (not necessarily orthogonal) such that \(P( {{\mathcal {K}}}) = {\mathcal {F}}\), see [3, 15]. This notion appears in connection to facial reduction algorithms and approaches for regularizing convex conic optimization problems. It turns out that projectional exposedness implies amenability [7, Proposition 9] and in dimension four or less, amenability implies projectional exposedness [8, Corollary 6.4].

So far, the largest classes of hyperbolicity cones which are known to be projectionally exposed are the symmetric cones ( [7, Proposition 33]) and the polyhedral cones ( [15, Corollary 3.4]). However, [8, Corollary 6.4] and Theorem 1.1 imply that all hyperbolicity cones in dimension four or less are projectionally exposed, which is a curious fact that, at this moment, does not seem to have an obvious algebraic explanation.