n satisfying the following two conditions: (1.2) λ1

where U :=[

u1| u2| · · · | un−1| un

], Λ := diag[λ1, λ2,· · · , λn−1, λn] with eigen-pairs (λ_i, u_i) for i = 1, 2,· · · , n satisfying the following two conditions:

(1.2) λ1 ≥ · · · ≥ λn−1 > 0 > λn, and u^T_i uj =

{ 1 if i = j, 0 if i̸= j.

Such cone has been previously studied in the literature [16,17]. More specifically, Stern and Wolkowicz [17] characterizes the conditions of the spectrum of a given real-valued symmetric matrix based on the existence of a corresponding ellipsoidal cone. They also provide an equivalent description on exponential nonnegativity for second-order cone [16], which is related to the solution set of a linear autonomous system ˙ξ = Aξ. On the application side, the ellipsoidal cone is further used to model rendezvous of the multiple agents system and measure dispersion in directional datasets, see [3, 15] for more details.

Indeed, the class of ellipsoidal cones, as an important prototype in closed convex cones, covers several practical instances used in real world. For instances, if we set

(1.3) Q =

[ I_n−1 0

0 −1

]

and u_n= e_n,

then the ellipsoidal coneK_E reduces to the second-order cone [5, 6]:

Kⁿ:={

(¯x, x_n)∈ IRⁿ⁻¹× IR | ∥¯x∥ ≤ xn

}. When Q and u_n are set as

(1.4) Q =

[ In−1 0 0 − tan²θ

]

and un= en, then the ellipsoidal coneK_E reduces to the circular cone [4, 19]:

Lθ:={

(¯x, xn)∈ IRⁿ⁻¹× IR | ∥¯x∥ ≤ xntan θ} When Q and un are taken as

(1.5) Q =

[ M^TM 0

0 −1

]

and u_n= e_n, then the ellipsoidal coneK_E reduces to the elliptic cone [2]:

Kⁿ_M :={

(¯x, x_n)∈ IRⁿ⁻¹× IR | ∥M ¯x∥ ≤ xn

}.

Here ∥ · ∥ denotes the standard Euclidean norm, In−1 means the identity matrix of order n− 1, θ ∈ (0,^π₂), M is any nonsingular matrix of order n− 1 and en is the n-th column vector of I_n. Hence, the ellipsoidal cone is a natural generalization of second-order cone, circular cone and elliptic cone, whose relations [13, Remark 1]

are depicted in Fig. 1.

Not only the ellipsoidal cone includes the well known second-order cone, circular cone and elliptic cone as special cases, as mentioned above; but also the ellipsoidal cone and the second-order cone can be converted to each other, see [13, Theorem 2.1 and Theorem 2.2] or Section 2. Through this bridge, the class of ellipsoidal cones connect symmetric cones and nonsymmetric cones. It is well known that symmetric cones can be dealt with under the Euclidean Jordan Algebra (EJA) [1, 4–6, 9, 10], whereas there is no unified framework for the category of nonsymmetric cones yet.

For symmetric cones, the self-duality and homogeneity are the two key components.

ON THE SELF-DUALITY AND HOMOGENEITY OF ELLIPSOIDAL CONES 1357

Figure 1. The relations among Kⁿ,Lθ,Kⁿ_M, and K_E.

In this paper, we address the issue regarding how to make an ellipsoidal cone to be self-dual and homogeneous¹. Our contributions can be summarized as follows: (a) We provide new easy-calculate inner products associated with ellipsoidal cone and its special cases circular cone and elliptic cone, which make them to be self-dual. (b) Following an easy-check procedure, we show that ellipsoidal cone is homogeneous, see the proof of Theorem 3.1 below for more details. In some sense, we believe that such constructive analysis will be crucial for finding an unified framework for some families of nonsymmetric cone in the future.

2. Preliminaries

In this section, we recall some background materials about the ellipsoidal cone, including its interior, dual cone, and its connection to second-order cone. Most of them can be found in [13], we only extract some for our subsequent needs.

In what follows, the dual of ellipsoidal cone K_E under the standard Euclidean inner product⟨·, ·⟩ defined on IRⁿ, denoted by (K_E)^∗_⟨·,·⟩, is defined as

(K_E)^∗_⟨·,·⟩ :={y ∈ IRⁿ| ⟨x, y⟩ ≥ 0, ∀x ∈ K_E} .

1While finalizing a first version of this work, the authors became aware of a more general observation made in G¨uler [12], based on characteristic function of a cone. Following the discussion in [12], we know that if E is isomorphic to a self-dual cone inRⁿ, then E is self-dual with respect to some inner product onRⁿ. However, how to set such new inner product is not an easy task. On the other hand, ellipsoidal cone can be viewed as the image of second-order cone under an invertible linear transformation, therefore is homogenous due to theories on homogeneous cones [7, 8, 18]. In contrast to these qualitative analysis, we provide a quantitative approach to show the self-duality and homogeneity of ellipsoidal cones, in which more features and observations are obtained in this constructive way.

For any given vector x∈ K_E, due to the orthogonal property of{ui}ⁿ_i=1, there exists a vector α := [α1,· · · , αn]^T ∈ IRⁿ such that x = U α, which implies

x^TQx = α^TU^TQU α = α^TΛα =

∑n i=1

λ_iα²_i and u^T_nx = u^T_n ( _n

∑

i=1

α_iu_i )

= α_n. Hence, the set K_E can be rewritten as follows:

K_E = U ∆_α where ∆_α:=

{

α∈ IRⁿ ∑ⁿ

i=1

λ_iα²_i ≤ 0, αn≥ 0 }

.

If we take λ_i = 1 for i = 1,· · · , n − 1 and λn = −1, the set ∆α reduces to the second-order cone,

(2.1) Kⁿ:=

{

α∈ IRⁿ

n−1

∑

i=1

α²_i ≤ α²n, α_n≥ 0 }

.

For any α ∈ ∆α, in light of the relation (1.2) for {λi}ⁿ_i=1, we have ∆_α = DKⁿ, where D is a n× n diagonal matrix of the form

(2.2) D := diag

[

(λ₁)^−1/2,· · · , (λn−1)^−1/2, (−λn)^−1/2 ]

.

In other words, we obtain K_E = U ∆α = U DKⁿ = TKⁿ, where T := U D is a nonsingular matrix in IR^n×n. For simplicity, we denote by|Λ| := diag [|λ1|, · · · , |λn|], and hence|Λ| = D⁻². Similarly, we also obtain (KE)^∗_⟨·,·⟩ = (T^T)⁻¹Kⁿ= U D⁻¹Kⁿ= T|Λ|Kⁿ and (K_E)^∗∗_⟨·,·⟩ =(

(T|Λ|)^T)₋₁

Kⁿ= TKⁿ=K_E.

The next theorem sums up the aforementioned relations and presents a reformu- lation of the dual cone (KE)^∗_⟨·,·⟩, whose proofs can be seen in [13, Theorem 2.1 and Theorem 2.2]. Such relations are also depicted in Fig. 2.

Theorem 2.1. LetK_E be an ellipsoidal cone given as in (1.1) andKⁿ be a second- order cone given as in (2.1). Then, the following relations hold.

(a): K_E = TKⁿ andKⁿ= T⁻¹K_E;

(b): (K_E)^∗_⟨·,·⟩ = T|Λ|Kⁿ and Kⁿ=|Λ|⁻¹T⁻¹(K_E)^∗_⟨·,·⟩; (c): (K_E)^∗_⟨·,·⟩ = T|Λ|T⁻¹K_E and (K_E)^∗∗_⟨·,·⟩ =K_E; (d): (K_E)^∗_⟨·,·⟩ ={

y∈ IRⁿ| y^TQ⁻¹y ≤ 0, u^T_ny ≥ 0} .

Remark 2.2. Applying Theorem 2.1(d), the duals of circular cone and elliptic cone under the standard Euclidean inner product⟨·, ·⟩, denoted by (Lθ)^∗_⟨·,·⟩ and (K_Mⁿ )^∗_⟨·,·⟩, are respectively given by

(Lθ)^∗_⟨·,·⟩:={

(¯yn−1, yn)∈ IRⁿ⁻¹× IR | ∥¯yn−1∥ ≤ yncot θ}

=L^π_2−θ, (KMⁿ )^∗_⟨·,·⟩ :=

{

(¯yn−1, yn)∈ IRⁿ⁻¹× IR (M⁻¹)T

¯

yn−1 ≤ yⁿ}

=Kⁿ_(M−1)^T, where θ ∈ (

0,^π₂)

and ¯y_n−1 := (y₁, y₂,· · · , yn−1)^T ∈ IRⁿ⁻¹. It follows from the above relations and Fig. 2 that circular cone, elliptic cone and ellipsoidal cone are obviously not self-dual under the standard Euclidean inner product, except their common special case, i.e., second-order cone.

ON THE SELF-DUALITY AND HOMOGENEITY OF ELLIPSOIDAL CONES 1359

Figure 2. The graphs of a 3-dimensional ellipsoidal cone, its dual cone under the Euclidean inner product and a 3-dimensional second- order cone.

For convenience, we denote by intK_E the interior of ellipsoidal cone. It follows from Theorem 2.1 and [14, Theorem 6.6] that intK_E = T (intKⁿ). This together with the definition of Kⁿ imply that intKⁿ := {

α∈ IRⁿ| α^TQnα < 0, e^T_nα > 0} , where the matrix Q_n is defined as in (1.3), i.e.,

(2.3) Q_n:=

[ I_n−1 0

0 −1

]

∈ IR^n×n.

Then, for any given x∈ KE and its corresponding vector α = T⁻¹x∈ Kⁿ, we obtain intK_E ={

x∈ IRⁿ(T⁻¹x)^TQ_n(T⁻¹x) < 0, e^T_nT⁻¹x > 0} . From the definition of T , we also achieve two useful relations as follows:

(T⁻¹)T

Q_nT⁻¹=(

D⁻¹U⁻¹)T

Q_nD⁻¹U⁻¹= U D⁻¹Q_nD⁻¹U^T = U ΛU^T = Q, e^T_nT⁻¹= e^T_n(U D)⁻¹= e^T_nD⁻¹U⁻¹= e^T_nD⁻¹U^T = (−λn)^1/2u^T_n.

With these, an explicit expression for the interior of K_E is displayed in the fol- lowing theorem.

Theorem 2.3. Let KE be an ellipsoidal cone given as in (1.1). Then, the interior of K_E can be expressed as

intK_E = {

x∈ IRⁿ(T⁻¹x)^TQn(T⁻¹x) < 0, e^T_nT⁻¹x > 0}

= {

x∈ IRⁿ| x^TQx < 0, u^T_nx > 0} .

Remark 2.4. Likewise, we conclude from Theorem 2.3 that the interior ofLθ and K_Mⁿ are respectively described by

intLθ :={

(¯x_n−1, x_n)∈ IRⁿ⁻¹× IR | ∥¯xn−1∥ < xntan θ} , intKMⁿ :={

(¯xn−1, xn)∈ IRⁿ⁻¹× IR | ∥M ¯xn−1∥ < xn

}.

where ¯xn−1 := (x1, x2,· · · , xn−1)^T ∈ IRⁿ⁻¹.

3. Main results

In this section, we first introduce a new inner product with respect to the ellip- soidal coneK_E as in (1.1), which enables that the ellipsoidal cone is self-dual under this new setting. Then, we establish the homogeneity of the class of ellipsoidal cones.

In the sequel, the new inner product of x, y∈ IRⁿ is defined as follows:

(3.1) ⟨x, y⟩KE := x^T|Q|y

∥Qun∥ =− 1 λn

(U^Tx)^T|Λ|(U^Ty).

From the orthogonal property of{ui}ⁿ_i=1, there exist α∈ IRⁿ and β ∈ IRⁿsuch that x = U α and y = U β. For simplicity, we write

Λ := diag[ ¯Λn−1, λn]∈ IR^n×n, ¯Λn−1:= diag[λ1, λ2,· · · , λn−1]∈ IR^{(n−1)×(n−1)}, α := [ ¯α^T_n−1, αn]^T ∈ IRⁿ, ¯αn−1 := [α1, α2,· · · , αn−1]^T ∈ IRⁿ⁻¹,

β := [ ¯β_n^T₋₁, βn]^T ∈ IRⁿ, ¯βn−1 := [β1, β2,· · · , βn−1]^T ∈ IRⁿ⁻¹ and ¯M is a diagonal matrix of order n− 1 with the i-th element √

λ_i/(−λn), where i = 1,· · · , n − 1. It is easy to verify that the matrix ¯M also satisfies the equation (3.2) Λ¯_n−1+ λ_nM¯^TM = 0¯

and the ellipsoidal cone KE can be recast as KE = UKⁿM¯, where KⁿM¯ is an elliptic cone with the parameter ¯M .

After these discussions, we are ready to show that the ellipsoidal cone is self-dual under the new inner product.

Theorem 3.1. Under the new inner product (3.1), the ellipsoidal cone KE given as in (1.1) is self-dual, i.e., (K_E)^∗_⟨·,·⟩

KE :={x ∈ IRⁿ| ⟨x, y⟩_KE ≥ 0, ∀y ∈ K_E} = K_E. Proof First, we show the inclusion KE ⊆ (KE)^∗_⟨·,·⟩

KE. Suppose that x∈ KE, we need to verify that x∈ (K_E)^∗_⟨·,·⟩

KE. For any y∈ K_E, due to the orthogonal of{ui}ⁿ_i=1 and the fact KE = UKⁿM¯, there exist α, β ∈ KMⁿ¯ such that x = U α and y = U β.

ON THE SELF-DUALITY AND HOMOGENEITY OF ELLIPSOIDAL CONES 1361

Hence, we obtain

⟨x, y⟩_KE

= − 1

λ_n(U^Tx)^T|Λ|(U^Ty)

= − 1 λn

α^T|Λ|β

=

[ α¯_n−1 αn

]T[ _¯

Λ_n−1 (−λⁿ) 0

0 1

] [ β¯_n−1 βn

]

=

[ α¯_n−1 αn

]T[ M¯^TM¯ 0

0 1

] [ β¯_n−1 βn

]

= (M ¯¯α_n−1)T (M ¯¯β_n−1)

+ α_nβ_n

≥ (M ¯¯αn−1)_T (M ¯¯βn−1)

+ ¯M ¯αn−1 · ¯M ¯βn−1

≥ (M ¯¯αn−1)_T (M ¯¯βn−1)

+(M ¯¯αn−1)_T(M ¯¯βn−1)

≥ 0,

where the first inequality follows from the fact α, β ∈ KMⁿ¯ and the second one is obtained by Cauchy-Schwartz inequality. The above inequality says that x ∈ (KE)^∗_⟨·,·⟩

KE and henceK ⊆ (KE)^∗_⟨·,·⟩

KE.

Next, we prove that the reverse inclusion is also valid. Suppose that x∈ (K_E)^∗_⟨·,·⟩

KE. Similar to the above arguments, there exist α∈ IRⁿsuch that x = U α. To proceed, we discuss two cases:

Case (a): If ¯αn−1 = 0, then we have x = αnun. It suffices to verify that αn ≥ 0.

To see this, we choose y = u_n, which gives y^TQy = u^T_nQun= u^T_n

( _n

∑

i=1

λiuiu^T_i )

un= λn< 0 and u^T_ny = u^T_nun= 1 > 0.

These imply that y = u_n ∈ K_E. Then, using definitions of (K_E)^∗_⟨·,·⟩

KE and (3.1) yields

⟨x, y⟩_KE

= − 1

λ_n(U^Tx)^T|Λ|(U^Ty)

= − 1 λn

α^T|Λ|β

= [ 0

α_n

]T [ M¯^TM¯ 0

0 1

] [ 0 1

]

= α_n≥ 0.

This shows α_n≥ 0 and hence x ∈ K_E. Thus, we have proved (K_E)^∗_⟨·,·⟩

KE ⊆ K_E. Case (b): If ¯α_n−1̸= 0, then we set y = Uβ, where β = (−¯α^T_n−1,∥ ¯M ¯α_n−1∥)^T ∈ KⁿM¯. From the relation KE = UKⁿM¯, we know that y ∈ KE. using the fact ⟨x, y⟩KE ≥ 0

and ¯αn−1 ̸= 0, we have

⟨x, y⟩_KE =(M ¯¯α_n−1)T (M ¯¯β_n−1)

+ α_nβ_n= ¯M ¯α_n−1 (α_n− ∥ ¯M ¯α_n−1∥)

≥ 0, which deduces that αn−∥ ¯M ¯αn−1∥ ≥ 0, i.e., α ∈ KⁿM¯ and x = U α∈ K_E. This shows (K_E)^∗_⟨·,·⟩

KE ⊆ K_E. 

Similar to Theorem 3.1, for any x := (¯x^T_n−1, xn)^T ∈ IRⁿ and y := (¯y^T_n−1, yn)^T ∈ IRⁿ, using (1.4) and (1.5), we define two types of inner products for the settings of circular cone and elliptic cone:

⟨x, y⟩_Lθ := cot²θ· ¯x^Tn−1y¯_n−1+ x_ny_n, (3.3)

⟨x, y⟩_Kⁿ_M := (M ¯xn−1)^T(M ¯yn−1) + xnyn. (3.4)

Corollary 3.2. Under the new inner products given as in (3.3) and (3.4), the circular coneLθ and the elliptic coneKⁿ_M are self-dual, respectively. In other words, we have

(Lθ)^∗_⟨·,·⟩

Lθ :={x ∈ IRⁿ| ⟨x, y⟩_Lθ ≥ 0, ∀y ∈ Lθ} = Lθ, (KⁿM)^∗_⟨·,·⟩

KnM

:={x ∈ IRⁿ| ⟨x, y⟩_Kⁿ_M ≥ 0, ∀y ∈ KⁿM} = KⁿM.

Next, we establish the homogeneity of the class of ellipsoidal cones, which is another main result of this paper.

Theorem 3.3. The ellipsoidal cone given as in (1.1) is homogeneous.

Proof Our proof is inspired by [9, Chap I, pages 7-8] and we complete the arguments by six steps.

Step 1: The interior of ellipsoidal coneK_E is given as intK_E ={

x∈ IRⁿ| x^T(−Q)x > 0, u^Tnx > 0} , which follows from Theorem 2.3.

Step 2: We introduce the corresponding bilinear form [x, y] := x^T(−Q)y and rewrite intKE as intKE ={

x∈ IRⁿ| [x, x] > 0, u^Tnx > 0} . Step 3: Now, we define

Ξ :={

P ∈ IR^n×n| P is nonsingular and P^TQP = Q} .

It is clear to see that Ξ is nonempty because I_n∈ Ξ. For any P1, P₂ ∈ Ξ, we have P1P2 ∈ Ξ due to

(P₁P₂)^TQ(P₁P₂) = P₂^T(P₁^TQP₁)P₂ = P₂^TQP₂ = Q,

where the last two equations follow from the facts P₁^TQP₁ = Q and P₂^TQP₂ = Q.

Moreover, we know P⁻¹ ∈ Ξ for any P ∈ Ξ by observing (P⁻¹)^TQ(P⁻¹) = (P Q⁻¹P^T)⁻¹=(

P (P^TQP )⁻¹P^T)₋₁

= Q.

To sum up, we have shown that the set Ξ is a subgroup of GL(IRⁿ) (the set of all nonsingular matrices of order n) and closed under matrix multiplication and inverse.

Step 4: We next construct a set of transformations as below (3.5) Ξ_KE :={

T AT⁻¹ ∈ IR^n×n| A is nonsingular, A^TQ_nA = Q_n and A_n,n > 0} ,

ON THE SELF-DUALITY AND HOMOGENEITY OF ELLIPSOIDAL CONES 1363

where T = U D, D is defined as in (2.2), An,nis the (n, n)-entry of A and the matrix Q_nis defined as in (2.3). We will show that this set Ξ_KE is indeed an automorphism group of intK_E, that is, Aut(intK_E) = Ξ_KE.

(i) If P ∈ ΞKE, then there exists a nonsingular matrix A ∈ IR^n×n such that P = T AT⁻¹, A^TQ_nA = Q_n, A_n,n > 0 and hence the inverse of A now equals to Q⁻¹_n A^TQ_n. Besides these relations, we obtain A⁻¹_n,n= A_n,n> 0 and (A⁻¹)^TQ_nA⁻¹= (AQ⁻¹_n A^T)⁻¹ = (A(A^TQ_nA)⁻¹A^T)⁻¹ = Q_n, which show that P⁻¹ = T A⁻¹T⁻¹ ∈ Ξ_KE. Moreover, let P1, P2 ∈ ΞKE, there exist two nonsingular matrices A1, A2 ∈ IR^n×n such that P_i = T A_iT⁻¹, A^T_iQ_nA_i = Q_n and (A_i)_n,n > 0 for i = 1, 2 and P1P2 = T A1A2T⁻¹. Due to the properties of Ai (i = 1, 2), we know that (A1A2)^TQn(A1A2) = Qn. Then, it follows from [11, Proposition 7.1 and Proposi- tion 7.6] that the matrix A in (3.5) has a polar decomposition of the form

(3.6) A =

[ S 0 0 1

] [ (In−1+ vv^T)^1/2 v

v^T c

] ,

where S is an orthogonal matrix of order n− 1 and c =√

∥v∥²+ 1, v∈ IRⁿ⁻¹. In particular, for the matrices A₁, A₂, there exist S₁, S₂, two orthogonal matrices of order n− 1, and c1, c2, v1, v2 such that

A1 =

[ S₁ 0 0 1

] [ (I_n−1+ v₁v^T₁)^1/2 v₁ v₁^T c₁

] , A2 =

[ S₂ 0 0 1

] [ (I_n−1+ v₂v^T₂)^1/2 v₂ v₂^T c₂

] ,

with c_i =√

∥vi∥²+ 1 for i = 1, 2. It is easy to verify that

(A₁A₂)_n,n

= v₁^TS₂v₂+ c₁c₂

≥ c1c₂− ∥v1∥ · ∥S2v₂∥

= c1c2− ∥v1∥ · ∥v2∥

≥ √

∥v1∥²+ 1·√

∥v2∥²+ 1− ∥v1∥ · ∥v2∥

> 0,

where the second equation uses the fact that S₂ is an orthogonal matrices of order n− 1. From these results, we also obtain P1P2 ∈ Ξ_KE. Hence, the set Ξ_KE is closed under matrix multiplication and inverse as well. Notice that D⁻¹Q_nD⁻¹ = Λ. For

any x∈ IRⁿand P ∈ Ξ_KE, we have [P x, P x]

= (P x)^TQ(P x)

= x^TP^TQP x

= x^T(U DAD⁻¹U^T)^T(U ΛU^T)(U DAD⁻¹U^T)x

= x^TU D⁻¹A^TDU^T(U D⁻¹QnD⁻¹U^T)U DAD⁻¹U^Tx

= x^TU D⁻¹A^TQ_nAD⁻¹U^Tx

= x^TU D⁻¹QnD⁻¹U^Tx

= x^TU ΛU^Tx

= x^TQx

= [x, x] > 0,

which shows that Ξ_KE ⊆ Ξ and hence Ξ_KE is a subgroup of Ξ.

(ii) On the other hand, for any x∈ intK_E, there exist a element α = ( ¯α^T_n−1, αn)^T ∈ intKⁿM¯ such that x = U α and u^T_nx = u^T_nU α = αn > 0. For any P ∈ ΞKE, due to the structure of A as in (3.6) and T = U D, we obtain

u^T_n(P x)

= u^T_nU DAD⁻¹U^TU α

= e^T_nDAD⁻¹α

= e^T_nD

[ S 0 0 1

] [ (In−1+ vv^T)^1/2 v

v^T c

] D⁻¹α

= e^T_n

[ ( ¯Λn−1)^−1/2 0 0 (−λn)^−1/2

] [ S 0 0 1

] (3.7)

·

[ (I_n−1+ vv^T)^1/2 v

v^T c

] [ ( ¯Λn−1)^1/2 0 0 (−λn)^1/2

] α

= v^TM ¯¯α_n−1+ c· αn

≥ −∥v∥ · ∥ ¯M αn−1∥ + cαn

= √

∥v∥²+ 1· αn− ∥v∥ · ∥ ¯M α_n−1∥, (3.8)

where the fifth equation is obtained from (3.2). If v = 0, then u^T_n(P x) ≥ αn > 0;

otherwise, i.e., v̸= 0, from the fact α = (¯α^Tn−1, α_n)^T ∈ intKⁿM¯, we obtain u^T_n(P x) >

∥v∥·(αn−∥ ¯M α_n−1∥) > 0. These relations together with (3.8) show that P x ∈ intK_E for any given x∈ intKE and P ∈ ΞKE.

From (i) and (ii), we claim that the set Ξ_KE is indeed an automorphism group of intKE, i.e., Aut(intKE) = Ξ_KE. In addition, it is clear that its dilation transforma- tion ˜Ξ_KE := η· ΞKE with η > 0 is an automorphism group of intKE, too.

Step 5: The set Ξ_KE is nonempty. For example, we have two choices of A as below.

(a) A =

[ S 0 0 1

]

, where S is an orthogonal matrix of order n− 1. This subclass can be deduced from (3.6) when we set v = 0.

ON THE SELF-DUALITY AND HOMOGENEITY OF ELLIPSOIDAL CONES 1365

(b) A = Ht, where Ht are the hyperbolic rotations:

Ht=



 cosh t 0 sinh t 0 In−2 0 sinh t 0 cosh t



 , t ≥ 0.

Here cosh t and sinh t are hyperbolic cosine and hyperbolic sine, which are respec- tively given by

cosh t = e^t+ e^−t

2 , sinh t = e^t− e^−t

2 .

We can easily verify that Ht satisfies the condition H_t^TQnHt= Qn and (Ht)n,n = cosh t > 0 for any t≥ 0.

Step 6: In order to show that ellipsoidal cone is homogeneous, we only need to verify that for any given x ∈ intK_E, there exists an element ˜P ∈ ˜Ξ_KE such that P u˜ _n = x for u_n ∈ intK_E, which means that we need to find a positive scalar η and P ∈ ΞKE such that x = η· P un. From the relation intKE = T (intKⁿ) and the structure of Ξ_KE, the above requirements are equivalent to the following conditions:

for any given α ∈ intKⁿ, there exist a positive scalar η and a element A ∈ IR^n×n such that A^TQ_nA = Q_n, A_nn > 0 and α = (−λn)^1/2η· Aen, where e_n is the n-th column of In and T en = (−λn)^−1/2un, which reduces to the case in [9, Chap I, pages 7-8]. Thus, there exists S an orthogonal matrix of order n− 1 such that A =

[ S 0 0 1

]

Htand η =

(α^T(−Qn)α

−λn

)_1/2

. Moreover, from the relation x = T α and T = U D, we obtain

x =

([x, x]

−λn

)_1/2 T

[ S 0 0 1

]

HtT⁻¹un,

which shows that the ellipsoidal cone is homogeneous.  As special cases of ellipsoidal cone, the homogeneities of circular cone and elliptic cone follow immediately by Theorem 3.3.

Corollary 3.4. The circular cone Lθ and the elliptic cone Kⁿ_M are also homoge- neous.

4. Concluding remarks

In this paper, through introducing the new inner product (3.1) with respect to the ellipsoidal cone as in (1.1), we show its corresponding structural properties such as the self-duality and homogeneity. At the same time, due to the relationK_E = TKⁿ and [9, Chap I, pages 7-8], two interesting things are observed:

• If we set Q =

[ In−1 0

0 −1

]

, U = I_n and u_n= e_n,

then the new inner product (3.1) reduces to the standard Euclidean inner product.

• The set Ξ_KE, the automorphism group of intK_E, can be characterized as the similarity transformation of the automorphism group of intKⁿ under the matrix parameter T . Let us denote the automorphism group of intKⁿas Ξ_Kⁿ :={A ∈ IR^n×n| A is nonsingular, A^TQ_nA = Q_n and A_n,n> 0},

where An,n is the (n, n)-entry of A and the matrix Qn is defined as in (2.3). The relation of the automorphism group between intK_E and intKⁿ is depicted as

where the notation “Aut(·)” denotes the automorphism group of the given set.

To sum up, we believe that the analysis used in this paper will pave a way to tackling with other unfamiliar closed convex cones appeared in real world.

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Manuscript received May 14, 2018 revised May 19, 2018

Y. Lu

School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China E-mail address: [email protected]

J-S. Chen

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: [email protected]