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Algebraic boundaries of convex semialgebraic sets
 Rainer Sinn^{1}Email author
https://doi.org/10.1186/s4068701500220
© Sinn; licensee Springer. 2015
 Received: 3 November 2014
 Accepted: 23 February 2015
 Published: 20 March 2015
Abstract
We study the algebraic boundary of a convex semialgebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semialgebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semialgebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.
2010 Mathematics Subject Classification: Primary: 52A99, 14N05, 14P10; Secondary: 51N35, 14Q15
Keywords
 Algebraic boundary
 Convex semialgebraic set
 Projective dual variety
Introduction
The algebraic boundary of a semialgebraic set is the smallest algebraic variety containing its boundary in the Euclidean topology. For a fulldimensional polytope \(\mathbb {R}^{n}\), it is the hyperplane arrangement associated to its facets which has been studied extensively in discrete geometry and complexity theory in linear programming [4]. The algebraic boundary of a convex set which is not a polytope has recently been considered in other special cases, most notably the convex hull of a variety by Ranestad and Sturmfels, cf. [11] and [12]. This class includes prominent families such as the moment matrices of probability distributions and the highly symmetric orbitopes. It does not include examples such as hyperbolicity cones and spectrahedra, which have received attention from applications of semidefinite programming in polynomial optimisation, see [2] and [19], and statistics of Gaussian graphical models, see [17].
First steps towards using the algebraic boundary of a spectrahedron for a complexity analysis of semidefinite programming have been taken by Nie et al. [9]. For semidefinite liftings of convex semialgebraic sets via Lasserre relaxations or theta body construction, the singularities of the algebraic boundary on the convex set give obstructions, cf. [6,8].
So algebraic boundaries are central objects in applications of algebraic geometry to convex optimisation and statistics. In this paper, we want to consider the class of all convex sets for which the algebraic boundary is an algebraic hypersurface: convex semialgebraic sets with nonempty interior. Our goal in this paper is to extend the study of the algebraic boundary of the convex hull of a variety started by Ranestad and Sturmfels in [11] and [12] to general convex semialgebraic sets. The most natural point of view in the general setting is via convex duality and its algebraic counterpart in projective algebraic geometry. The first main theorem generalises and implies the correspondence between facets of a polytope with vertices of its dual polytope.
Theorem (Corollary 3.4).
Let \(C\subset \mathbb {R}^{n}\) be a compact convex semialgebraic set with 0∈int(C). Let Z be an irreducible component of the Zariski closure of the set of extreme points of its dual convex body. Then, the variety dual to Z is an irreducible component of the algebraic boundary of C.
For polytopes, this theorem is the whole story. In the general semialgebraic case, not every irreducible component of the algebraic boundary of C arises in this way, as we will see below. We study the exceptional cases and give a complete semialgebraic description of the exceptional families of extreme points in terms of convex duality (normal cones) and a computational way of getting a list of potentially exceptional strata from the algebraic boundary of the dual. This proves an assertion made by Sturmfels and Uhler in [17] Proposition 2.4.
The main techniques come from the duality theories in convex and projective algebraic geometry. For an introduction to convex duality, we refer to Barvinok’s textbook [1]. The duality theory for projective algebraic varieties is developed in several places, e.g. Harris [7], Tevelev [18] or GelfandKapranovZelevinsky [5].
This article is organised as follows: In Section ‘The algebraic boundary and convexity’, we introduce the algebraic boundary of a semialgebraic set and discuss some special features of convex semialgebraic sets coming from their algebraic boundary. The section sets the technical foundation for Section ‘The algebraic boundary of convex semialgebraic sets’, where we prove the main results of this work.
The algebraic boundary and convexity
This section is supposed to be introductory. We will fix the notation and observe some basic features of convex semialgebraic sets, their algebraic boundary and some special features relying on this algebraic structure. The main results will be proven in the following section.
Definition 2.1.
Let \(S\subset \mathbb {R}^{n}\) be a semialgebraic set. The algebraic boundary of S, denoted as ∂ _{ a } S, is the Zariski closure in \(\mathbb {A}^{n}\) of the Euclidean boundary of S.
Remark 2.2.
In this paper, we fix a subfield k of the complex numbers. The most important choices to have in mind are the reals, the complex numbers or the rationals. When we say Zariski closure, we mean with respect to the kZariski topology, i.e. the topology on \(\mathbb {C}^{n}\) (resp. \(\mathbb {P}(\mathbb {C}^{n+1})\)) whose closed sets are the algebraic sets defined by polynomials (resp. homogeneous polynomials) with coefficients in k. The set \(\mathbb {C}^{n}\) (resp. \(\mathbb {P}(\mathbb {C}^{n+1})\)) equipped with the kZariski topology is usually denoted \({\mathbb {A}^{n}_{k}}\) (resp. \({\mathbb {P}^{n}_{k}}\)). We drop the field k in our notation. The statements in this paper are true over any subfield k of the complex numbers given that the semialgebraic set in consideration can be defined by polynomial inequalities with coefficients in \(k\cap \mathbb {R}\).
If we are interested in symbolic computation, we tend to consider semialgebraic sets defined by polynomial inequalities with coefficients in and take Zariski closures in the Zariski topology.
We first want to establish that the algebraic boundary of a convex body is a hypersurface.
Definition 2.3.
A subset of \(\mathbb {R}^{n}\) is called regular if it is contained in the closure (in the Euclidean topology) of its interior.
Remark 2.4.
Every convex semialgebraic set with nonempty interior is regular, and the complement of a convex semialgebraic set is also regular.
Lemma 2.5.
Let \(\emptyset \neq S\subset \mathbb {R}^{n}\) be a regular semialgebraic set and suppose that its complement \(\mathbb {R}^{n}\setminus S\) is also regular and nonempty. Each irreducible component of the algebraic boundary of S has codimension 1 in \(\mathbb {A}^{n}\), i.e. ∂ _{ a } S is a hypersurface.
Proof.
Example 2.6.
The assumption of S being regular cannot be dropped in the above lemma. Write \(h:=x^{2}+y^{2}+z^{2}1\in \mathbb {R}[x,y,z]\). Let S be the union of the unit ball with the first coordinate axis, i.e. \(S = \{(x,y,z)\in \mathbb {R}^{3}\colon y^{2}h(x,y,z)\leq 0,z^{2}h(x,y,z)\leq 0\}\). The algebraic boundary of S is the union of the sphere \(\mathcal {V}(h)\) and the line \(\mathcal {V}(y,z)\), which is a variety of codimension 1 with a lower dimensional irreducible component.
Remark 2.7.
In the above proof of Lemma 2.5, we argue over the field of real numbers. The algebraic boundary of S, where the Zariski closure is taken with respect to the kZariski topology for a different field k, is also a hypersurface. It is defined by the reduced product of the Galois conjugates of the polynomial defining ∂ _{ a } S over , whose coefficients are algebraic numbers over k.
Corollary 2.8.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set with nonempty interior. Its algebraic boundary is a hypersurface.
This property characterises the semialgebraic compact convex sets.
Proposition 2.9.
A compact convex set with nonempty interior is semialgebraic if and only if its algebraic boundary is a hypersurface.
Proof.
The converse follows from results in semialgebraic geometry. Namely if the algebraic boundary ∂ _{ a } C is an algebraic hypersurface, its complement \(\mathbb {R}^{n}\setminus (\partial _{a} C)(\mathbb {R})\) is a semialgebraic set and the closed convex set C is the closure of the union of finitely many of its connected components. This is semialgebraic by BochnakCosteRoy [3] Proposition 2.2.2 and Theorem 2.4.5. □
By the construction of homogenisation in convexity, the algebraic boundary of a pointed and closed convex cone relates to the algebraic boundary of a compact base via the notion of affine cones in algebraic geometry.
Remark 2.10.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set, and let \(\text {co}(C)\subset \mathbb {R}\times \mathbb {R}^{n}\) be the convex cone over C embedded at height 1, i.e. co(C)={(λ,λ x):λ≥0,x∈C}. Since a point (1,x) lies in the boundary of co(C) if and only if x is a boundary point of C, the affine cone \(\left \{(\lambda,\lambda x)\colon \lambda \in \mathbb {C}, x\in \partial _{a} C\right \}\) over the algebraic boundary of C is a constructible subset of the algebraic boundary of co(C). More precisely, we mean that \(\partial _{a} \text {co}(C)= \widehat {X}\), where X is the projective closure of ∂ _{ a } C with respect to the embedding \(\mathbb {A}^{n}\mapsto \mathbb {P}^{n}\), (x _{1},…,x _{ n })↦(1:x _{1}:…:x _{ n }) and \(\widehat {X}\) is the cone over the projective variety X, i.e. all points \(x\in \mathbb {A}^{n+1}\) such that the line through x and the origin is in X.
Recall that a closed convex cone \(C\subset \mathbb {R}^{n}\) is called pointed if C∩(−C)={0}, i.e. it does not contain a line.
Corollary 2.11.
We will now take a look at convex duality for semialgebraic sets. Given a compact convex set \(C\subset \mathbb {R}^{n}\), we write \(C^{o} = \{\ell \in (\mathbb {R}^{n})^{\ast }\colon \ell (x)\geq 1 \text {for all} x\in C\}\) for the dual convex set. We use the notation X _{reg} for the set of all regular (or smooth) points of an algebraic variety X.
Proposition 2.12.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set with 0∈int(C), and set S:=∂ C ^{ o }∩X _{reg}(∂ _{ a } C ^{ o }). For every ℓ∈S, the face supported by ℓ is a point. The set S is an open and dense (in the Euclidean topology) semialgebraic subset of the set ∂ C ^{ o } of all supporting hyperplanes to C.
Proof.
Let ℓ be in S and denote by ev_{ x } the point evaluation of linear functionals at \(x\in \mathbb {R}^{n}\). Given \(x\in \mathbb {R}^{n}\) such that ev_{ x } defines a supporting hyperplane to C ^{ o } at ℓ, then ℓ(x)=−1 and C ^{ o } lies in one halfspace defined by ev_{ x }. Therefore, \(\left (\partial _{a} C^{o}\right)(\mathbb {R})\) lies locally around ℓ in one halfspace defined by ev_{ x } and so ev_{ x } defines the unique tangent hyperplane to ∂ _{ a } C ^{ o } at ℓ. Now, we show that x is an extreme point of C, exposed by ℓ. Suppose \(x=\frac 12 (y+z)\) with y,z∈C, then ℓ(y)=−1 and ℓ(z)=−1. Since y and z are, by the same argument as above, also normal vectors to the tangent hyperplane T_{ ℓ } ∂ _{ a } C ^{ o }, we conclude x=y=z.
The set S is open in ∂ C ^{ o }, because \((\partial _{a} C^{o})_{\text {reg}}(\mathbb {R})\) is open in the Euclidean topology. It is also dense in ∂ C ^{ o } because for every ℓ∈C ^{ o } at least one irreducible component of ∂ _{ a } C ^{ o } has full local dimension at ℓ. □
The same statement is true for convex cones: We denote the dual convex cone to \(C\subset \mathbb {R}^{n+1}\) as \(C^{\vee } = \left \{\ell \in (\mathbb {R}^{n+1})^{\ast }\colon \ell (x)\geq 0 \text { for all }x\in C\right \}\).
Corollary 2.13.
Let \(C\subset \mathbb {R}^{n+1}\) be a pointed closed semialgebraic convex cone with nonempty interior and set S:=∂ C ^{∨}∩(∂ _{ a } C ^{∨})_{reg}. For every ℓ∈S, the face supported by ℓ is an extreme ray of C. The set S is open and dense (in the Euclidean topology) semialgebraic subset of ∂ C ^{∨}. □
Example 2.14.
(a) In the case that C is a polytope, the set S of regular points of the algebraic boundary is exactly the set of linear functionals exposing extreme points. Indeed, in this case the algebraic boundary of C is a union of affine hyperplanes, namely the affine span of its facets. A point in ∂ C is a regular point of the algebraic boundary ∂ _{ a } C if and only if it lies in the relative interior of a facet, cf. Barvinok [1] Theorem VI.1.3. These points expose the vertices of C ^{ o }.
In this example, the set S=∂ C ^{ o }∩(∂ _{ a } C ^{ o })_{reg} in the above corollary is the boundary of C ^{ o } except for the six intersection points of the irreducible components of ∂ _{ a } C ^{ o } on the boundary of the dual convex set.
The extreme points (resp. rays) of a convex set play an important role for duality. They will also be essential in a description of the algebraic boundary using the algebraic duality theory. So, we fix the following notation:
Definition 2.15.
(a) Let \(C\subset \mathbb {R}^{n}\) be a convex semialgebraic set. We denote by Ex_{ a }(C) the Zariski closure of the union of all extreme points of C in \(\mathbb {A}^{n}\).
(b) Let \(C\subset \mathbb {R}^{n+1}\) be a semialgebraic convex cone. We write Exr_{ a }(C) for the Zariski closure of the union of all extreme rays of C in \(\mathbb {A}^{n+1}\).
Remark 2.16.
(a) Note that the union of all extreme points of a convex semialgebraic set is a semialgebraic set by quantifier elimination because the definition is expressible as a firstorder formula in the language of ordered rings, cf. BochnakCosteRoy [3] Proposition 2.2.4. Therefore, its Zariski closure is an algebraic variety whose dimension is equal to the dimension of Ex(C) as a semialgebraic set, cf. BochnakCosteRoy [3] Proposition 2.8.2. Of course, the same is true for convex cones and the Zariski closure of the union of all extreme rays.
Lemma 2.17.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set with 0∈int(C). For a general extreme point x∈Ex_{ a }(C), there is a supporting hyperplane ℓ _{0}∈∂ C ^{ o } exposing the face x and a semialgebraic neighbourhood U of ℓ _{0} in ∂ C ^{ o } such that every ℓ∈U supports C in an extreme point x _{ ℓ } and all x _{ ℓ } lie on the same irreducible component of Ex_{ a }(C) as x.
By general, we mean in this context that the statement is true for all points in a dense (in the Zariski topology) semialgebraic subset of Ex_{ a }(C).
Proof.
By Straszewicz’s Theorem (see Rockafellar [14] Theorem 18.6) and the Curve Selection Lemma from semialgebraic geometry (see BochnakCosteRoy [3] Theorem 2.5.5), a general extreme point is exposed. Let y∈Ex(C) be an exposed extreme point contained in a unique irreducible component Z of Ex_{ a }(C) and denote by ℓ _{ y } an exposing linear functional. Let Z _{1},…,Z _{ r } be the irreducible components of Ex_{ a }(C) labelled such that Z=Z _{1}. Since the sets Z _{ i }∩∂ C⊂C are closed, they are compact. Now, ℓ _{ y } is strictly greater than −1 on Z _{ i }∩∂ C for i>1 and therefore, there is a neighbourhood U in ∂ C ^{ o } of ℓ _{ y } such that every ℓ∈U is still strictly greater than −1 on Z _{ i }∩∂ C. The intersection of this neighbourhood with the semialgebraic set S of linear functionals exposing extreme points, which is open and dense in ∂ C ^{ o } in the Euclidean topology by Proposition 2.12, is nonempty and open in ∂ C ^{ o }. Pick ℓ _{0} from this open set, then the extreme point x exposed by ℓ _{0} has the claimed properties. □
Example 2.18.
(a) Again, the above lemma has a simple geometric meaning in the case of polytopes: Every extreme point of the polytope is exposed exactly by the relative interior points of the facet of the dual polytope dual to it, again by Barvinok [1] Theorem VI.1.3.
(b) In Example 2.14(b), the boundary of the convex set C consists of extreme points and a single onedimensional face. So the only linear functional not exposing an extreme point of C is the dual face to the edge of C, which is (0,−1)∈Ex(C ^{ o }). As seen in Figure 1, the extreme points of C ^{ o } that support C in an extreme point are the ones on the quadric irreducible components with positive y coordinate. They come in two irreducible families dual to the two parabolas in the algebraic boundary of C.
By homogenisation, we can prove the analogous version of the above lemma for closed and pointed convex cones.
Corollary 2.19.
Let \(C\subset \mathbb {R}^{n+1}\) be a pointed closed semialgebraic convex cone with nonempty interior. Let F _{0}⊂C be an extreme ray of C such that the line [F _{0}] is a general point of \(\mathbb {P}\text {Exr}_{a}(C)\). Let Z be the irreducible component of \(\mathbb {P}\text {Exr}_{a}(C)\) with [F _{0}]∈Z. Then, there is a supporting hyperplane ℓ _{0}∈∂ C ^{∨} exposing F _{0} and a semialgebraic neighbourhood U of ℓ _{0} in ∂ C ^{∨} such that every ℓ∈U supports C in an extreme ray F _{ ℓ } of C contained in the regular locus of Z, i.e. [F _{ ℓ }]∈Z _{reg}. □
The above notion of the general now translates into the projective notion, i.e. the statement is true for points in a dense semialgebraic subset of the semialgebraic set of extreme rays as a subset of \(\mathbb {P}\text {Exr}_{a}(C)\subset \mathbb {P}^{n}\).
The algebraic boundary of convex semialgebraic sets
We now consider projective dual varieties: Given an algebraic variety \(X\subset \mathbb {P}^{n}\), the dual variety \(X^{\ast }\subset (\mathbb {P}^{n})^{\ast }\) is the Zariski closure of the set of all hyperplanes \([H]\in (\mathbb {P}^{n})^{\ast }\) such that H contains the tangent space to X at some regular point x∈X _{reg}. For computational aspects of projective duality, we refer to RanestadSturmfels [11] and RostalskiSturmfels [15].
Proposition 3.1.
Proof.
Let \(Y\subset \mathbb {P}\partial _{a} C\) be an irreducible component of the algebraic boundary of C. Let \(x\in \widehat {Y}\cap \partial C\) be a general point and \(H\subset \mathbb {R}^{n+1}\) be a supporting hyperplane to C at x. We argue similarly to the proof of Proposition 2.12: Since C lies in one halfspace defined by H, so does \(\widehat {Y}\) locally around x. Therefore, H is the tangent hyperplane \(\mathrm {T}_{x}\widehat {Y}\). Now, the tangent hyperplane to \(\widehat {Y}\) at x is unique, because \(\widehat {Y}\) has codimension 1. So the set of all supporting hyperplanes to C at x is an extreme ray of the dual convex cone. Since \(\widehat {Y}\cap C\) is Zariski dense in \(\widehat {Y}\), the hyperplanes tangent to \(\widehat {Y}\) at points \(x\in \widehat {Y}\cap C\) are dense in the dual variety to Y. □
Remark 3.2.
Let Z⊂Exr_{ a }(C) be an irreducible component. Then, the dual variety to \(\mathbb {P} Z\subset \mathbb {P}^{n}\) is a hypersurface in \((\mathbb {P}^{n})^{\ast }\), which follows from the biduality theorem in projective algebraic geometry Tevelev [18] Theorem 1.12, because \(\mathbb {P} Z\) cannot contain a dense subset of projective linear spaces of dimension ≥1. Suppose \(\mathbb {P} Z\) contained a dense subset of projective linear spaces of dimension ≥1, then the set Z∩Exr(C), which is dense in Z, would contain a Zariski dense subset of an affine linear space of dimension at least 2. This contradicts the fact that the set of extreme rays Exr(C) does not contain any line segments other than those lying on the rays themselves.
In the language of cones, our first main theorem is the following.
Theorem 3.3.
Proof.
Let \(\mathbb {P} Z\subset \mathbb {P}\text {Exr}_{a}(C)\) be an irreducible component of the locus of extreme rays of C. By Corollary 2.19, a general extreme ray \([\!F_{0}]\in \mathbb {P} Z\cap (\mathbb {P} \text {Exr}(C))\) is exposed by ℓ _{0}∈∂ C ^{∨} and there is a semialgebraic neighbourhood U of ℓ _{0} in ∂ C ^{∨} such that every ℓ∈U exposes an extreme ray F _{ ℓ } of C such that \([F_{\ell }]\in (\mathbb {P} Z)_{\text {reg}}\). The hyperplane \(\mathbb {P}\text {ker}(\ell)\) is tangent to \(\mathbb {P} Z\) at [F _{ ℓ }] because \(\mathbb {P} Z\) is locally contained in C; so \(\mathbb {P} U\) is a semialgebraic subset of \(\mathbb {P} Z^{\ast }\) of full dimension and the claim follows. □
In the Section ‘Introduction’, we gave an affine version of the preceding theorem that follows from it via homogenisation.
Corollary 3.4.
Let \(C\subset \mathbb {R}^{n}\) be a compact convex semialgebraic set with 0∈int(C). Let Z be an irreducible component of the Zariski closure of the set of extreme points of its dual convex body. Then, the variety dual to Z is an irreducible component of the algebraic boundary of C. More precisely, the dual variety to the projective closure \(\overline {Z}\) of Z with respect to the embedding \(\mathbb {A}^{n}\to (\mathbb {P}^{n})^{\ast }\), x↦(1:x) is an irreducible component of the projective closure of ∂ _{ a } C with respect to \(\mathbb {A}^{n}\to \mathbb {P}^{n}\), x↦(1:x).
Proof.
We homogenise the convex body and its dual convex body by embedding both at height 1 to get convex cones \(\text {co}(C)=\left \{(\lambda,\lambda x)\colon \lambda \geq 0, x\in C\right \}\subset \mathbb {R}\times \mathbb {R}^{n}\) and \(\text {co}(C^{o}) = \left (\text {co}(C)\right)^{\vee }\subset \mathbb {R}\times \left (\mathbb {R}^{n}\right)^{\ast }\). The projective closure \(\overline {Z}\) of the irreducible component Z⊂Ex_{ a }(C ^{ o }) with respect to the embedding \(\mathbb {A}^{n}\to \left (\mathbb {P}^{n}\right)^{\ast }\), x↦(1:x) is an irreducible component of \(\mathbb {P} \text {Exr}_{a}(\text {co}(C)^{\vee })\). By the above Theorem 3.3, the dual variety to \(\overline {Z}\) is an irreducible component of \(\mathbb {P} (\partial _{a} \text {co}(C))\), which is the projective closure of an irreducible component of the algebraic boundary of C with respect to the embedding \(\mathbb {A}^{n}\to \mathbb {P}^{n}\), x↦(1:x). □
Corollary 3.5.
Let \(C\subset \mathbb {R}^{n+1}\) be a pointed closed semialgebraic convex cone with nonempty interior. We have \((\mathbb {P} \partial _{a} C)^{\ast } =\mathbb {P} \text {Exr}_{a}(C^{\vee })\). □
Remark 3.6.
The following statement gives a complete semialgebraic characterisation of the irreducible subvarieties Y⊂Exr_{ a }(C) with the property that Y ^{∗} is an irreducible component of the algebraic boundary of C ^{∨}.
Theorem 3.7.
Proof.
This says that Σ is dense in \(\text {CN}(\mathbb {P}Z)\), i.e. \(\dim (\Sigma)=\dim (\text {CN}(\mathbb {P}Z))+2=n+1\) if and only if \(\mathbb {P}Z^{\ast }\) is an irreducible component of \(\mathbb {P} \partial _{a} C^{\vee }\).
Remark 3.8.
We want to compare this theorem to the result of Ranestad and Sturmfels in [11]: They consider the convex hull of a smooth algebraic variety \(X\subset \mathbb {P}^{n}\) and make the technical assumption that only finitely many hyperplanes are tangent to the variety X in infinitely many points, which is needed for a dimension count in the proof. We get rid of this technical assumption in the above theorem. The assumption that the extreme rays are Zariski dense in the variety Z in question compares best to the RanestadSturmfels assumption. It is semialgebraic in nature. As an example, consider the cone of positive semidefinite real symmetric 3×3 matrices: It is the convex hull of the rank 1 matrices, which are the Veronese embedding \(X = v_{2}(\mathbb {P}^{2}) \subset \mathbb {P}^{5}\). This cone is selfdual from the point of convex geometry. In this case, there are infinitely many hyerplanes that are tangent to X at infinitely many points, namely every rank 1 matrix is tangent to X along a conic, when interpreted as a hyperplane via the trace inner product on the space of symmetric matrices. To see that, note that the tangent space to X at x x ^{ t } is the set of all matrices of the form y x ^{ t }+x y ^{ t } for \(y\in \mathbb {P}^{2}\) and a matrix A is perpendicular to that tangent space with respect to the trace inner product if and only A x=0. So given that rk(A)=1, the hyperplane will be tangent to X along a plane conic, namely v _{2}(ker(A))⊂X. Our result shows that the algebraic boundary of the dual convex cone is the dual variety to X, which is the determinantal hypersurface.
The corresponding affine statement to Theorem 3.7 is the following. We take projective closures with respect to the same embeddings as in the affine version Corollary 3.4 of Theorem 3.3 above.
Corollary 3.9.
Proof.
Again, the proof is simply by homogenising as above. Note that the dimension of the normal cone does not change when homogenising. □
In the following affine examples, we will drop the technical precision of taking projective closures and talk about the dual variety to an affine variety to make them more readable.
Example 3.10.

The extreme points of the dual convex set are dense in X _{ i } via \(\mathbb {R}^{n}\to \left (\mathbb {P}^{n}\right)^{\ast }\), x↦(1:x).

A general extreme point of the dual convex set in X _{ i } exposes a face of C of dimension codim(X _{ i })−1.
and we see both conditions in action. First, the dual variety to the affine line x=1 is (−1,0), which is not an extreme point of C ^{ o }. The first condition mentioned above shows that the line \(\mathcal {V}(x1)\) corresponding to the second inequality in the first description is not an irreducible component of ∂ _{ a } C. In the second description, the dual variety to the affine line \(y = \sqrt {2}(x1)\) is the point \(P = \left (1,\frac {1}{\sqrt {2}}\right)\), which is an extreme point of C ^{ o }. The normal cone \(\phantom {\dot {i}\!}N_{C^{o}}(\{P\})\) is onedimensional, because the supporting hyperplane is uniquely determined  it is the bitangent \(\mathcal {V}(x+1)\) to the quartic. So by the second condition above, the line \(\mathcal {V}(y\sqrt {2}(x1))\) is not an irreducible component of ∂ _{ a } C.
Corollary 3.11.

\(\overline {Y}^{\ast }\subset Z_{\text {sing}}[Z]\) or

\(\overline {Y}^{\ast }\) is contained in the algebraic boundary of the semialgebraic subset Ex(C)∩Z of Z.
Proof.
Let \(Z\subset \overline {\text {Ex}_{a}(C)}\) be an irreducible subvariety. If \(\ell \in \left (\mathbb {R}^{n}\right)^{\ast }\) defines a supporting hyperplane to an extreme point x∈Ex(C) that is an interior point of the semialgebraic set Ex(C)∩Z as a subset of Z and (1:x)∈Z _{reg}, then the variety Z lies locally in one of the half spaces defined by (1:ℓ) and therefore (1:ℓ) is tangent to Z at (1:x). In particular, the dimension of the normal cone N _{ C }({x}) is bounded by the local codimension of Z at (1:x). Now if \(\overline {Y}^{\ast }\) is strictly contained in Z, it cannot contain (1:x) by Corollary 3.9 because \(\dim \left (\overline {Y}^{\ast }\right)< \dim (Z)\). □
The set Z∩Ex(C) in the above corollary does not need to be a regular semialgebraic set. So the second condition can also occur in the following way.
Example 3.12.
Consider the convex hull C of the half ball \(\left \{(x,y,z)\in \mathbb {R}^{3}\colon x^{2}+y^{2}\right.\,+\left. z^{2}\leq 1, x\geq 0\right \}\) and the circle \(X = \left \{(x,y,z)\in \mathbb {R}^{3}\colon x^{2}+y^{2}\leq 1, z=0\right \}\). The Zariski closure of the extreme points of C is the sphere S ^{2}. Every point of the circle X is a regular point of S ^{2} and X is contained in the algebraic boundary of Ex(C)∩S ^{2}⊂S ^{2}, because the semialgebraic set Ex(C)∩S ^{2} does not have local dimension 2 at the extreme points (x,y,0)∈X∩Ex(C) where x<0. The algebraic boundary of the dual convex set has three irreducible components, namely the sphere S ^{2} and the dual varieties to the two irreducible components X and \(\mathcal {V}\left (y^{2}+z^{2}1,x\right)\) of ∂ _{ a }(Ex(C)∩S ^{2})⊂S ^{2}. Note that X and \(\mathcal {V}\left (y^{2}+z^{2}1,x\right)\) are not irreducible components of Ex_{ a }(C)=S ^{2}, so that we are looking at varieties of the second type in the above Corollary. The reader may find a picture of C and its polar on the author’s website, see [16].
The following examples show how the statement of the corollary can be used to determine the algebraic boundary in concrete cases.
Example 3.13.
computed in RostalskiSturmfels [15] Section 1.1, Equations 1.7 and 1.8. The first quartic is the dual variety to the quartic \(\mathcal {V}(\det (Q))\). The two quadric hypersurfaces are products of linear forms over , and they are the dual varieties to the four corners of the pillow, namely \(\frac {1}{\sqrt {2}}(1,1,1)\), \(\frac {1}{\sqrt {2}}(1,1,1)\), \(\frac {1}{\sqrt {2}}(1,1,1)\) and \(\frac {1}{\sqrt {2}}(1,1,1)\). These four points are extreme points of P and singular points of \(\mathcal {V}(\det (Q))\).
Another interesting consequence of Corollary 3.9 concerns the semialgebraic set Ex(C).
Corollary 3.14.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set with 0∈int(C). Every extreme point x of C is a central point of the dual variety \(\overline {Y}^{\ast }\) of at least one irreducible component \(\overline {Y}\) of \(\overline {\partial _{a} C^{o}}\) via \(\mathbb {A}^{n}\to \mathbb {P}^{n}\), x↦(1:x).
A point x on a real algebraic variety \(X\subset \mathbb {P}^{n}\) is called central if \(X(\mathbb {R})\) has full local dimension around x. Equivalently, x∈X is central if it is the limit of a sequence regular real points of X, cf. BochnakCosteRoy [3] Section 7.6 and Proposition 10.2.4.
Proof.
We take a short look at implications of this corollary to hyperbolicity cones.
Example 3.15.
The assumption on the hyperbolicity cone being smooth is essential: Consider the hyperbolicity cone of \(p = y^{2}z(x+z)(xz)^{2}\in \mathbb {R}[x,y,z]\) with respect to (0,0,1). The cubic \(\mathcal {V}(p)\subset \mathbb {R}^{3}\) is singular along the line \(\mathbb {R}(1,0,1)\) and the algebraic boundary of the dual convex cone has an additional irreducible component, namely the hyperplane dual to this line because the normal cone has dimension 2 at this extreme ray, see Figure 3.
How can we compute these exceptional varieties of extreme points? Given the algebraic boundary of the dual convex set, the following theorem gives an answer. In its statement, we use an iterated singular locus: The kth iterated singular locus of a variety X, denoted by X _{ k,sing}, is the singular locus of the (k−1) iterated singular locus. The first iterated singular locus is the usual singular locus of X.
Theorem 3.16.
Let \(C\subset \mathbb {R}^{n}\) be a compact semialgebraic convex set with 0∈int(C), and suppose that every point ℓ∈∂ C ^{ o } is a regular point on every irreducible component of ∂ _{ a } C ^{ o } containing it. Let Z⊂Ex_{ a }(C ^{ o }) be an irreducible subvariety such that \(\overline {Z}^{\ast }\) is an irreducible component of \(\overline {\partial _{a} C}\). If codim(Z)=1, then Z is an irreducible component of ∂ _{ a } C ^{ o }. If codim(Z)=c>1, then Z is an irreducible component of an iterated singular locus, namely it is an irreducible component of one of the varieties (∂ _{ a } C ^{ o })_{sing},(∂ _{ a } C ^{ o })_{2,sing},…,(∂ _{ a } C ^{ o })_{ c−1,sing}.
Proof.
Assume codim(Z)=c>1, and let ℓ∈Z∩Ex(C ^{ o }) be a general point. Since Whitney’s condition (a) is satisfied for (X _{reg},Z) at ℓ for every irreducible component X⊂∂ _{ a } C ^{ o } with Z⊂X by BochnakCosteRoy [3] Theorem 9.7.5, every extreme ray \(\mathbb {R}_{+} x\) of \(\phantom {\dot {i}\!}N_{C^{o}}(\{\ell \})\) is tangent to Z at ℓ by Corollary 3.14. Since the extreme rays of the normal cone \(\phantom {\dot {i}\!}N_{C^{o}}(\{\ell \})\) span the smallest linear space containing it, the dimension of Z is bounded from above by \(\text {codim}\left (N_{C^{o}}(\{\ell \})\right)\). The assumption that \(\overline {Z}^{\ast }\) is an irreducible component of \(\overline {\partial _{a} C}\) implies \(\dim (Z) = \text {codim}(N_{C^{o}}(\{\ell \}))\) by Corollary 3.9. It follows that the tangent space T_{ ℓ } Z is the lineality space of the convex cone \(\phantom {\dot {i}\!}N_{C^{o}}(\{\ell \})^{\vee }\). To show that Z is an irreducible component of (∂ _{ a } C ^{ o })_{ j,sing}, suppose Y⊂(∂ _{ a } C ^{ o })_{ k,sing} is an irreducible component with \(Z\subsetneq Y\) and Y _{reg}∩Z≠∅ and let ℓ∈Z∩Ex(C ^{ o }) be a general point with ℓ∈Y _{reg}. Then, \(\mathrm {T}_{\ell } Z\subsetneq \mathrm {T}_{\ell } Y\) and there is an extreme ray \(\mathbb {R}_{+} x\) of \(\phantom {\dot {i}\!}N_{C^{o}}(\ell)\) with x∈Ex(C) and \(\phantom {\dot {i}\!}x\vert _{\mathrm {T}_{\ell } Y} \neq 0\). By Corollary 3.14, there is an irreducible component X⊂∂ _{ a } C ^{ o } such that x is a central point of X ^{∗}. So by assumption, ℓ∈X _{reg} and x∈(T_{ ℓ } X)^{⊥}. Since \(\phantom {\dot {i}\!}x\vert _{\mathrm {T}_{\ell } Y}\neq 0\), the varieties Y and X intersect transverally at ℓ. So, \(Z\subset Y\cap X \subsetneq Y\) and Y∩X⊂(∂ _{ a } C ^{ o })_{ j,sing} are irreducible components for some j>k because the multiplicity of a point in Y∩X in ∂ _{ a } C ^{ o } is higher than the multiplicity of a general point on Y. Induction on the codimension of Z proves the theorem. □
Remark 3.17.
(a) This theorem gives a computational way to get a list of candidates for the dual varieties to irreducible components of the algebraic boundary of C, given the algebraic boundary of C ^{ o }. Certain of these candidates may fail to contribute an irreducible component due to semialgebraic constraints. For illustration, we will apply it to two examples.
(b) The assumption that all irreducible components of ∂ _{ a } C ^{ o } are smooth along the boundary of C ^{ o } is used to show that the stratification into iterated singular loci is sufficient in this case. In general, it may be necessary to refine this stratification such that Whitney’s condition (a) is satisfied for all adjacent strata, see Example 3.20.
Example 3.18 (cf. Remark 3.6).
We consider the convex set \(C\subset \mathbb {R}^{2}\) in the plane defined by the two inequalities x ^{2}+y ^{2}≤1 and x≤3/5, see Figure 2. Its algebraic boundary is the plane curve \(\mathcal {V}((x^{2}+y^{2}1)(x3/5))\). The dual convex body is the convex hull of the set \(\{(X,Y)\in \mathbb {R}^{2} \colon X^{2}+Y^{2}\leq 1, X\geq 3/5\}\) and the point (−5/3,0). Its algebraic boundary is the curve \(\partial _{a} C^{o} = \mathcal {V}((X^{2}+Y^{2}1)(4Y3X5)(4Y+3X+5))\). Its three irreducible components are smooth, and its singular locus consists of three points, namely (−5/3,0) and (−3/5,±4/5). By the above theorem, a complete list of candidates for the algebraic boundary of C are the dual varieties to the circle \(\mathcal {V}(X^{2}+Y^{2}1)\) and the irreducible components of the first iterated singular locus, i.e. the lines dual to the points (−5/3,0) and (−3/5,±4/5). In fact, the last two points do not contribute an irreducible component to ∂ _{ a } C, because the normal cone to C ^{ o } at these points is onedimensional, cf. Corollary 3.9.
We can also look at it dually and compute the algebraic boundary ∂ _{ a } C ^{ o } from the singularities of the algebraic boundary of C: The curve ∂ _{ a } C is reducible, all components are smooth and its singular locus consists of two points, namely (3/5,±4/5). Both of these points dualize to irreducible components of ∂ _{ a } C ^{ o }.
Example 3.19.
Viewed dually, this example is more complicated. The algebraic boundary of C ^{ o } is the surface of degree 8 defined by the above polynomial, which has singularities along the boundary of C ^{ o }. So the above theorem is not applicable in this case but the conclusion is still true and we compute the iterated singular loci for demonstration. In conclusion, we will find the dual varieties of the cylinders, which are circles, as irreducible components of the singular locus of ∂ _{ a } C ^{0}. They are the two irreducible components of Ex_{ a }(C ^{ o }):
The singular locus of the surface has 4 irreducible components: the dual varieties to the cylinders, which are circles, namely \(\mathcal {V}(Z,X^{2}+Y^{2}1)\) and \(\mathcal {V}(X,4Y^{2}+4Z^{2}4Y3)\), a complex conjugate pair of quadrics \(\mathcal {V}(2 Y^{2}Y+2, 4 X^{2}3 Z^{2}2 Y Z^{2}+8 Y4)\), and a curve of degree 12, which we denote by X _{12}. The second iterated singular locus, which is the singular locus of the union of these four irreducible curves, consists of 24 points. Sixteen of them are the singular points of X _{12}, and the other eight points are intersection points of X _{12} with the complex conjugate pair of quadrics \(\mathcal {V}(2 Y^{2}Y+2, 4 X^{2}3 Z^{2}2 Y Z^{2}+8 Y4)\). The two circles dual to the cylinders intersect the curve X _{12} only in singular points of the latter. There are no other intersection points of the irreducible components of (∂ _{ a } C ^{ o })_{ sing }. Of these 24 points in (∂ _{ a } C ^{ o })_{2,sing}, only four are real. They are \((\pm \sqrt {5/9},2/3,0)\) and \((0,1/6,\pm \sqrt {5/9})\). Now, the difficult job is to exclude those varieties that do not contribute irreducible components to the algebraic boundary of C. The dual variety to ∂ _{ a } C ^{ o } is only a curve, so it cannot be an irreducible component of ∂ _{ a } C. Next, we discuss the irreducible components of (∂ _{ a } C ^{ o })_{ sing }: The dual varieties to the complex conjugate pair of quadrics cannot be an irreducible component of ∂ _{ a } C either, because the real points will not be dense in this hypersurface. Why the dual variety to the curve X _{12} is not an irreducible component of ∂ _{ a } C is not obvious. Of the irreducible components of (∂ _{ a } C ^{ o })_{2,sing}, the four real points must be considered as potential candidates for dual varieties to irreducible components of ∂ _{ a } C.
To close, we want to consider an example of a convex set whose algebraic boundary is not smooth along its Euclidean boundary and for which the conclusion of the Theorem 3.16 is false. As remarked above, the stratification into iterated singular loci must be refined to a stratification that is Whitney (a)regular.
Example 3.20.
Note that Whitney’s condition (a) for \((\mathcal {V}(g),\mathcal {V}(x1,y))\) is not satisfied at p because a hyperplane that is in limiting position for supporting hyperplanes to the teardrop C ^{′} does not contain the line \(\mathcal {V}(x1,y)\). Refining the stratification of iterated singular loci into a Whitney (a)regular stratification would detect this special extreme point.
Declarations
Acknowledgements
This work is part of my PhD thesis. I would like to thank my advisor Claus Scheiderer for his encouragement and support, the Studienstiftung des deutschen Volkes for their financial and ideal support of my PhD project and the National Institute of Mathematical Sciences in Korea, which hosted me when I finished this paper.
Authors’ Affiliations
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