Abstract
Let K be an ndimensional equiangular cone in a Euclidean vector space. Let \(0<\phi <\pi \) be the common angle between any pair of vectors taken among the generators of the cone. We obtain an explicit formula for the maximum angle of K in terms of n and \(\phi \). We explain also how to identify a pair \(\{u,v\}\) of unit vectors in K achieving the maximum angle. Finally, we analyze a counterintuitive phenomenon that occurs when n is large: the equiangular cone K can be constructed so that u and v are almost opposite, whereas \(\phi \) is near \(\pi /2\).
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1 Introduction
Let E be a Euclidean vector space of dimension \(d\ge 3\) and unit sphere \(S_E\). Let K be a nonzero closed convex cone in E. The dimension of K is defined as the dimension of the linear span of K. To avoid trivialities, we assume that \(n:=\textrm{dim}(K)\) is at least three. The maximum angle of K is defined as
where \(\langle \cdot , \cdot \rangle \) is the inner product of E and \(\Vert \cdot \Vert \) is the associated norm. A pair \(\{u, v\}\) of unit vectors in K achieving the maximum angle is called an antipodal pair of K. The term \(\theta _\textrm{max}(K)\) arises in a great variety of applications, cf. Clarke et al. [1], Peña and Renegar [10], and Iusem and Seeger [5], just to mention a few publications. Unless K is highly structured and has a lot of symmetry in it, evaluating the optimal value of the nonconvex optimization problem (1) is a difficult task. Easily computable formulas for \(\theta _\textrm{max}(K)\) are given in Gourion and Seeger [3, Proposition 2] for Schur cones, in Seeger [12, Example 5.8]) for p norm cones, and in Iusem and Seeger [7, Theorem 1] for ellipsoidal cones. A challenging problem still unsolved is that of characterizing the maximum angle of the cone of copositive symmetric matrices of a prescribed order, cf. [2, 4]. The case of a pointed polyhedral cone has been treated in a number of references, but only from an algorithmic point of view, cf. [3, 6, 11]. This note examines the case in which K is an n dimensional equiangular cone of aperture parameter \(0<\phi <\pi \). This means that
for some set \(\{g_1,\ldots , g_n\}\) of linearly independent vectors in E such that
The unit vectors \(\{g_1,\ldots , g_n\}\) are called the generators of K. Equiangular cones are pointed polyhedral cones. They are not necessarily full dimensional cones, because n could be smaller than the dimension of the underlying space E. Condition (3) says that all pairs \(\{g_i,g_j\}\) of generators form the same angle, namely, \(\phi \). As observed in Iusem and Seeger [8, Lemma 3], the common angle \(\phi \) and the number of generators n are bound by the inequality
The reason behind (4) is that if \(G: {\mathbb {R}}^n\rightarrow E\) is the linear map given by \(Gx:= \sum _{j=1}^nx_jg_j\) and \(G^\top : E\rightarrow {\mathbb {R}}^n\) is the adjoint linear map, then the Gramian matrix \(G^\top G \) is positive definite and therefore its determinant
is positive. Inequality (4) implies that \(\phi _n:= \arccos (1/(n1))\) is an upper bound for the aperture parameter of an ndimensional equiangular cone. The next result is easy, but we record it for the sake of completeness.
Theorem 1
Let K be an ndimensional equiangular cone of aperture parameter \(\phi \le \pi /2\). Then \(\theta _\textrm{max}(K)= \phi \). In particular, any pair of generators achieves the maximum angle of K.
Recall that a generator of a pointed polyhedral cone K is a unit vector \(w\in K\) whose associated ray \(\{tw:t\ge 0\}\) is a onedimensional face of the cone. The polyhedral cone K considered in Theorem 1 is nonobtuse in the sense that \(\langle u,v\rangle \ge 0\) for all \(u, v\in K\). For nonobtuse polyhedral cones, the maximum angle is attained by at least one pair of generators, cf. [6, Proposition 6.2]. This is in essence the proof of Theorem 1. The case \(\phi >\pi /2\) is much more difficult to handle, because in such a context the maximum angle of K is no longer attained by a pair of generators. The next theorem is in the spirit of Theorem 1, but it concerns the case of an obtuse angle \(\phi \). Theorem 2, a cornerstone of this note, will be stated and proven in the next section.
2 Obtuse counterpart of Theorem 1
The proof of Theorem 2 relies on three lemmas. The first one has to do with the intrinsic dual cone of an equiangular cone. The dual cone of a closed convex cone K in E is given by
whereas the intrinsic dual cone of K is given by \( K^\circledast := K^*\cap \textrm{span}(K)\). Of course, both dual cones coincide if K spans the entire space E. We recall also that a closed convex cone K in E is simplicial if K admits a basis of E as a set of generators.
Lemma 1
Let K be an ndimensional equiangular cone of aperture parameter \(0<\phi <\pi \). Then \(K^\circledast \) is an ndimensional equiangular cone of aperture parameter
Proof
Let K be as in (2)(3). We consider \(L:=\textrm{span}(K)\) as a Euclidean vector space equipped with the same inner product as E. Since \(\{g_1,\ldots , g_n\}\) is a basis of the vector space L, the linear map \(G:{\mathbb {R}}^n\rightarrow L\) is a bijection and
where \(\{h_1,\ldots , h_n\}\) is yet another basis of L. Note that K and \(K^\circledast \) are mutually dual simplicial cones in L. There is no loss of generality in assuming that the \(h_i\)’s are unit vectors. We now explain how to construct such vectors. For ease of presentation, we identify the ndimensional vector space L with \({\mathbb {R}}^n\) and view G as a square matrix of order n. Let \(F:=G^{\top }\) be the transpose of the inverse matrix of G and let \(H=[h_1,\ldots , h_n]\) be the matrix obtained by normalizing the columns of \(F=[f_1,\ldots f_n]\). Formula (6) follows from a standard calculus rule for the dual cone of a simplicial cone in \({\mathbb {R}}^n\). To check that (6) is an equiangular cone, we apply a proof technique used in Seeger and Torki [13, Proposition 2.3]. Firstly, we write the equiangularity hypothesis (3) as
where \(c:= \cos \phi \), \(I_n\) is the identity matrix of order n, and \(\textbf{1}_n\) is the ndimensional column vector of ones. By passing to the inverse matrix on each side of (7), we get
The latter equality yields
for all \(i\in {\mathbb {N}}_n:=\{1,\ldots ,n\}\). In particular, the \(f_i\)’s are of equal length. By (4), the constant \(\kappa \) is well defined and positive. Furthermore, \(\langle h_i, h_j\rangle = (1/\kappa )\langle f_i, f_j\rangle =  (1+(n2)c)^{1}c \) for all \(i\not =j\). This proves that \(K^\circledast \) is an ndimensional equiangular cone of parameter (5). If we do not wish to identify L and \({\mathbb {R}}^n\), then \(G^{\top }\) is to be understood as the adjoint linear map of \(G^{1}: L\rightarrow {\mathbb {R}}^n\). The proof follows the same steps as before, because \(G^\top G: {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) can still be viewed as a matrix of order n. \(\square \)
Thanks to the binding relation (4), the quotient inside arccosinus in (5) is well defined and belongs to the open interval \(]1,1[\). The transformation \(\phi \mapsto \phi ^*\) is clearly an involution, that is to say, \((\phi ^*)^*= \phi \). In other words, we can invert (5) and write
This observation is consistent with the fact that K is the intrinsic dual cone of \(K^\circledast \). Note that \(\cos \phi \) and \(\cos \phi ^*\) are of different signs (unless \(\phi =\phi ^*= \pi /2\)). Parenthetically, (5) and (8) can be written in the more symmetric form
We baptize this identity as the reflection law for equiangular cones.
The second lemma is more sophisticated. It is a result due to Iusem and Seeger [8] and concerns the angular spectrum of an acute equiangular cone. We need to recall some terminology. A nonzero closed convex cone K is acute if \(\langle u,v\rangle >0 \) for all \(u, v\in K\backslash \{0\}\). A critical pair of K is a pair \(\{u,v\}\) of distinct unit vectors in K satisfying
The system (9) is obtained by writing down the KarushKuhnTucker optimality conditions for the nonconvex optimization problem (1). The angle formed by a critical pair is called a critical angle of K. The angular spectrum or set of critical angles of K is denoted by \(\textrm{AS}(K)\).
Lemma 2
Let K be an ndimensional equiangular cone of aperture parameter \(\phi <\pi /2\). Then
with
Up to minor cosmetic changes, Lemma 2 corresponds to Theorem 5 in [8]. The third and last lemma is a general duality result for angular spectra established in [9]. We formulate below a particular version of [9, Theorem 3] that is well adapted to the context of this note.
Lemma 3
Let K be a pointed polyhedral cone in E. Then
In Lemma 3 it is implicitly assumed that K is of dimension at least three. Formula (11) remains true even beyond a polyhedral context. We are assuming polyhedrality because in such a case the angular spectrum \(\textrm{AS}(K)\) has finitely many elements. The largest element is \(\theta _\textrm{max}(K)\) and the smallest one is denoted by \(\theta _\textrm{min}(K)\). For obvious reasons, we say that \(\theta _\textrm{min}(K)\) is the minimum angle of K. Without further ado we state our much announced result.
Theorem 2
Let K be an ndimensional equiangular cone of aperture parameter \(\phi >\pi /2\). Then
with \(k:= \lfloor n/2\rfloor \) being the lower integer part of n/2.
Proof
Lemma 1 says that \(K^\circledast \) is an ndimensional equiangular cone of parameter \(\phi ^*\) given by (5). Since \(\phi > \pi /2\), the term \(c:=\cos \phi \) is negative and
is positive. Hence, \(\phi ^*= \arccos c^*\) is smaller than \(\pi /2\) and we can apply Lemma 1 to \(K^\circledast \). We get a formula similar to (10) for the set \(\textrm{AS}(K^\circledast )\), changing of course \(\phi \) to \(\phi ^*\). If we apply Lemma 3 to \(K^\circledast \), then we get
Since \(\theta _\textrm{max}(K)\) is the largest element of \( \textrm{AS}(K)\), we obtain
where \((p^*, q^*)\) is a solution to the integer programming problem
Since \(c^*\) is positive, minimizing
is the same task as maximizing
For a fixed p, the ratio (16) increases with q. Hence, the constraint \(p+q\le n\) is active at any solution (p, q) to (15). We must then maximize
with respect to \(p\in \{1,\ldots , n 1\}\). A quick examination of this univariate integer programming problem shows that the maximum of \(\zeta (p)\) is attained at \(p^*=k:=\lfloor n/2 \rfloor \). To complete the proof of (12), we substitute \(( p^*, q^*) =(k, nk) \) into (14) and carry out a short algebraic manipulation, namely, we exploit the trigonometric identity \( \pi \arccos (t)= \arccos (t)\) and use (13) to return from \(c_*\) to the original variable \(\phi \). \(\square \)
As said before, the smallest element of \(\textrm{AS}(K)\) is called the minimum angle of K. The minimum angle of a cone is undoubtedly less popular than the maximum angle, but it also has a certain number of applications, cf. [9, p. 24]. Since \(\theta _\textrm{min}(K)= \pi  \theta _\textrm{max}(K^\circledast )\) holds true for any pointed polyhedral cone K of dimension at least three, the first part of the next corollary comes for free as a combination of Theorem 1 and Theorem 2. The formula for the cardinality of \(\textrm{AS}(K)\) is obtained by using [8, Corollary 4] and the proof technique of Theorem 2.
Corollary 1
Let K be an ndimensional equiangular cone of aperture parameter \(\phi \). Then
where \(k:=\lfloor n/2\rfloor \).
We end this section by mentioning yet another straightforward consequence of Theorem 2. Let \(\Theta (n, \phi )\) be the maximum angle of an ndimensional equiangular cone of aperture parameter \(\phi \). While considering the term \(\Theta (n,\phi )\), it is tacitly understood that \(n\ge 3\) and \(0<\phi <\phi _n\). Note that \(\Theta (n, \phi )\) does not depend on a specific choice of K. From Theorem 1 and Theorem 2, we know that
By a continuity argument, one may expect \(\Theta (n, \phi ) \phi \) to be small if \(\phi \) is slightly obtuse. It would be strange for instance to have \(\Theta (n, \phi )\) near \(\pi \) when \(\phi \) is near \(\pi /2\). This counterintuitive situation occurs however when n is large.
Corollary 2
For all \(\varepsilon >0\), there exists an equiangular simplicial cone K that is almost unpointed and whose aperture parameter is slightly obtuse:
3 Further comments
Equiangular cones form a rather narrow class of convex cones. However, such cones are of importance for several reasons:

As said before, there are only few classes of convex cones for which it is possible to derive an explicit formula for the maximum angle. By Theorem 1 and Theorem 2, the class of equiangular cones falls into this category. For equiangular cones it is also possible to get an explicit formula for the angular spectrum and for the cardinality of such a set, cf. Corollary 1. Estimating the cardinality of the angular spectrum of a general polyhedral cone is computationally expensive.

Axial symmetry is a useful property in the theory of convex cones. Equiangular cones are not axially symmetric, but they have symmetry rank equal to one, cf. [13, Section 6] for the proof of this result and for the definition of the symmetry rank of a cone. Being of symmetry rank equal to one is not as strong as being axially symmetric, but it is enough for a number of practical issues (for instance, for the determination of a “central” axis).

Equiangular cones admit an explicit formula for the socalled least partial volume, cf. [14, Proposition 2.3]. The numerical computation of the least partial volume of a general polyhedral cone is awfully difficult. So, it pays to ask for equiangularity.

A Reuleaux triangle, which is an example of a planar strictly convex compact set of constant width, is constructed starting from an equilateral triangle. Analogously, an equiangular cone can be used as an ingredient for constructing a strictly convex cone of constant opening angle. This theme is being explored by J.P. Moreno (Univ. Autónoma de Madrid) and the second author in a working paper entitled “Angular structure of Reuleaux cones”. That a proper cone K is of constant opening angle means that the opening angle
$$\begin{aligned} {\mathfrak {f}}_K(u):= \max _{v\in K,\,\Vert v\Vert =1} \arccos \,\langle u,v\rangle \end{aligned}$$of K relative to a unit vector u remains constant if u moves on the boundary of K. Being of constant opening angle is a fairly restrictive property, so one should not expect to embrace a large class of cones. Things are as they are.
3.1 Techniques for constructing equiangular cones
Given \(n\ge 3\) and \(0<\phi <\phi _n\), how to construct an ndimensional equiangular simplicial cone of aperture parameter \(\phi \)? Although this question is not difficult, for the benefit of the reader we indicate two different construction methods. Note that \(\phi _n:= \arccos (1/(n1))\) decreases from \(2\pi /3\) to \(\pi /2\) as n goes from 3 to infinity. Hence, if n is large, then \(\phi \) must be nonobtuse or just slightly obtuse. For instance, if \(n=100\), then \(\phi \) must be smaller than \(\arccos (1/99)\approx 0.5032\,\pi \) (roughly, 90.6 degrees). Our first construction method boils down to solving the nonlinear equation
in the variable \(t\in {\mathbb {R}}\). Consider the matrix G(t) whose columns are the unit vectors
where \(e_k\) is the kth canonical vector of \({\mathbb {R}}^n\). The determinant formula
shows that the columns of G(t) are linearly independent, except when \(t\in \{1n,1\}\). On the other hand, a quick computation shows that
for all \(i\not =j\). Hence, for all \(t\in {\mathbb {R}}\backslash \{n1, 1\}\), the set \(K(t):= \textrm{cone}\{g_1(t), \ldots , g_n(t)\}\) is an ndimensional equiangular simplicial cone of aperture parameter
Since \(\gamma \) is a continuous function and decreases from \(\phi _n\) to 0 as t goes from \(1n\) to 1, the nonlinear Eq. (17) has a unique solution in the open interval \(]1n, 1[\). Furthermore, such a solution can be expressed as a function of n and \(\phi \) as indicated below:
The second expression in (19) is one of the two solutions to the quadratic equation
obtained by passing to cosine on each side of (17). The cone \(K(t_{n,\phi })\) constructed in this way is an ndimensional equiangular simplicial cone of aperture parameter \(\phi \).
A second construction technique was suggested to us by a referee, to whom we express our appreciation. We identify E with the space \({\mathbb {R}}^n\) and form a regular simplex \(R:= \textrm{co}\{v_1,\ldots , v_n\} \) in the plane \(\{x\in {\mathbb {R}}^n:x_n=0\}\). Constructing a regular \((n1)\)dimensional simplex offers no difficulty. By using a translation and a dilation if necessary, we may assume that R admits the origin of \({\mathbb {R}}^n\) as centroid and that \(v_1,\ldots , v_n\) are unit vectors. In such a case, \(\langle v_i, v_j\rangle = 1/(n1)\) when \(i\not = j\). We now lift each \(v_i\) to the hyperplane \(\{x\in {\mathbb {R}}^n:x_n=t\}\), where t is a positive parameter. After normalization, we get the unit vectors \(g_i:= [1+t^2]^{1/2} (v_i+ te_n)\) for \(i\in {\mathbb {N}}_n\), where \(e_n\) is the nth canonical vector of \({\mathbb {R}}^n\). The polyhedral cone generated by the \(g_i\)’s is an equiangular simplicial cone with aperture parameter
Note that \(\psi (t)\) decreases continuously from \(\phi _n\) to 0 as t goes from 0 to infinity. So, if we wish to get an aperture parameter equal to \(\phi \), then we simply chose
Remark 1
Let \(E_1\) and \(E_2\) be Euclidean spaces of the same dimension. Let \(K_1\) and \( K_2\) be closed convex cones in \(E_1\) and \(E_2\), respectively. One says that \(K_1\) and \(K_2\) are isometrically equivalent if \(K_2\) is the image of \(K_1\) under some linear isometry \(L:E_1\rightarrow E_2\). Two equiangular cones of equal dimension and aperture parameter are necessarily isometrically equivalent. In view of this observation, one can always assume that E is the usual Euclidean space \({\mathbb {R}}^n\).
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Gourion, D., Seeger, A. Angular structure of equiangular cones in Euclidean spaces. Aequat. Math. 98, 1225–1234 (2024). https://doi.org/10.1007/s00010023009947
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DOI: https://doi.org/10.1007/s00010023009947