Circular cone convexity and some inequalities associated with circular cones

R E S E A R C H Open Access

Circular cone convexity and some inequalities associated with circular cones

Jinchuan Zhou¹, Jein-Shan Chen^2*and Hao-Feng Hung²

*Correspondence:

jschen@math.ntnu.edu.tw

2Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

Full list of author information is available at the end of the article

Abstract

The study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity offby exploiting the second-order differentiability off^Lθ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angleθ, which gives us a new insight when looking into properties about circular cone.

MSC: 26A27; 26B05; 26B35; 49J52; 90C33; 65K05 Keywords: circular cone; convexity; determinant; trace

1 Introduction

Recently, much attention has been paid to the nonsymmetric cone optimization problems, see [–] and the references therein. Unlike symmetric cones [], there is no unified struc- ture for nonsymmetric cones. Hence, how to tackle nonsymmetric cone optimization is still an issue. For symmetric cone optimization, the algebraic structure associated with symmetric cones, including second-order cone and positive semi-definite matrix cones, allows us to study them via exploiting the unified Euclidean Jordan algebra []. In gen- eral, the way to deal with nonsymmetric cone optimization depends on the feature of the associated nonsymmetric cone. In this paper, we focus on a special nonsymmetric cone, circular coneLθ. The circular cone [–] is a pointed closed convex cone having hyper- spherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation. Let its half-aperture angle beθ withθ∈(, ^◦). Then, it is mathematically expressed as

Lθ:=

x= (x,x)^T∈R×R^n–|x≥ xcosθ

=

x= (x,x)^T∈R×R^n–|x≥ xcotθ .

Real applications of a circular cone lie in some engineering problems, for example, in the formulation for optimal grasping manipulation for multi-fingered robots, the grasping force ofith finger is subject to a circular cone constraint, see [, ] and references for more details.

AlthoughLθis a nonsymmetric cone, we can, due to its special structure, establish the explicit form of orthogonal decomposition (or spectral decomposition) [] as

x=λ_(x)·u^()_x +λ_(x)·u^()_x , ()

©2013Zhou et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

where

⎧⎨

⎩

λ_(x) =x–xcotθ, λ_(x) =x_+x_tanθ and

⎧⎪

⎨

⎪⎩

u^()x =_+cot^θ

 

cotθIn–

 –¯x =

sin^θ –(sinθcosθ)x¯ , u^()x =_+tan^θ

 

tanθIn–



¯ x =

cos^θ (sinθcosθ)¯x

withx¯=x/xifx= , andx¯being any vectorwinR^n–satisfyingw=  ifx= .

Clearly,x∈Lθif and only ifλ_(x)≥.

The formula () allows us to define the following vector-valued function:

f^Lθ(x) :=f λ_(x)

u^()_x +f λ_(x)

u^()_x , ()

wheref is a real-valued function fromJtoRwithJbeing a subset inR. LetSbe the set of allx∈Rⁿwhose spectral valuesλ_i(x) fori= ,  belong toJ,i.e.,S:={x∈Rⁿ|λ_i(x)∈J,i=

, }. According to [], we know thatSis open if and only ifJis open. In addition, asJis an interval, thenSis convex because

min

λ_(x),λ_(y)

≤λ_

βx+ ( –β)y

≤λ_

βx+ ( –β)y

≤max

λ_(x),λ_(y)

, ∀β∈[, ].

Throughout this paper, we always assume thatJ is an interval inR. Clearly, asθ= ^◦, L^◦reduces to the second-order cone and the above expressions () and () correspond to the spectral decomposition and the SOC-function associated with the second-order cone, respectively (see [, ] for more information regardingf^soc).

It is well known that in dealing with symmetric cone optimization problems, such as second-order cone optimization problems and positive semi-definite optimization prob- lems, this type of vector-valued functions plays an essential role. Inspired by this, we study the properties off^Lθ, which is crucial for circular cone optimization problems. In our previous works, we have studied the smooth and nonsmooth analysis off^Lθ [, ];

and the circular cone monotonicity and second-order differentiability off^Lθ []. From the aforementioned research, there is an interesting observation: some properties com- monly shared byf^socandf^Lθ are independent of the angleθ; for example,f^Lθ is direc- tionally differentiable, Fréchet differentiable, semi-smooth if and only iff is directionally differentiable, Fréchet differentiable, semi-smooth; while some properties are dependent on the angleθ; for example,f^Lθ withf(t) = –/tfort>  is circular cone monotone as θ∈[^◦, ^◦), but not circular cone monotone asθ∈(, ^◦).

In this paper, we further study the circular cone convexity off. More precisely, a real- valued functionf:J→Ris said to beLθ-convex of ordernonSif for anyx,y∈S,

f^Lθ

βx+ ( –β)y

Lθ βf^Lθ(x) + ( –β)f^Lθ(y), ∀β∈[, ].

The characterization ofLθ-convexity is based on the observation thatfisLθ-convex if and only if (f^Lθ)(x)(h,h)∈Lθfor allh∈Rⁿ. Our result shows that the circular cone convexity requires that the angleθ belongs in [^◦, ^◦). In particular, we show thatf isLθ-convex of order  if and only ifθ∈[^◦, ^◦) andf is convex.

On the other hand, using the spectral decomposition (), we define thedeterminantand traceofxin the framework of circular cone as

det(x) :=λ(x)λ(x) and tr(x) :=λ(x) +λ(x),

respectively. In the symmetric cone setting, the concepts of determinant and trace are the key ingredients of barrier and penalty functions which are used in barrier and penalty methods (including interior point methods) for symmetric cone optimization, see [–

]. Here we further study some basic inequalities ofdet(x) andtr(x) in the framework of circular cone. As seen in Section , the obtained inequalities are classified into three categories: (i) the first class is independent of the angle (i.e., still holds in the framework of circular cone); (ii) the second class is dependent on the angle, for example, forx,y∈Lθ, the inequality

det(e+x+y)≤det(e+x)det(e+y),

wheree= (, , . . . , )∈Rⁿ, fails asθ ∈(, ^◦) but holds asθ ∈[^◦, ^◦); (iii) the third class always fails no matter what value ofθ is chosen. These results give us a new insight into a circular cone and make us focus more on the role played by the angleθ.

The notation used in this paper is standard. For example, denote by Rⁿ the n- dimensional Euclidean space and by R+ the set of all nonnegative real scalars, i.e., R+={t∈R|t≥}. For x,y∈Rⁿ, the inner product is denoted byx^Ty. LetSⁿ mean the spaces of all real symmetric matrices inR^n×n, and letSⁿ+denote the cone of positive semi-definite matrices. We writexLθ yto stand forx–y∈Lθ. Finally, we define ^_:=  for convenience.

2 Circular cone convexity

The main purpose of this section is to provide characterizations ofLθ-convex functions.

First, we need the following technical lemma.

Lemma . Givenα_i∈Rfor i= , . . . , andβ_i∈Rfor i= , , ,we define

F(β,β,β) :=αβ_^+αβ_^+αβ_^β_^+αβ_^β_^+αβ_^β_^+αβββ_^. () IfF(β_,β_,β_)≥for all(β_,β_,β_)∈R^,then

α_≥, α_≥, α_≥, α_≥, α_≥–√ α_α_. Furthermore,if

α^_≤

⎧⎨

⎩

αα_ forα_≥,

[α_– (α^_/α_)]α_ forα_∈[–√

α_α_, ), ()

thenF(β,β_,β_)≥for all(β,β_,β_)∈R^.

Proof If β_ = , thenF(β,β_,β_) =β_^[αβ_^+α_β_^]. From F(β,β_,β_)≥, we have α_β_^+α_β_^≥. Thus,α_≥ by lettingβ_→ andα_≥ by lettingβ_→.

Ifβ_= , thenF(β_,β_,β_) =β_^[α_β_^+α_β_^]. FromF(β_,β_,β_)≥, we obtainα_≥ andα_≥.

Ifβ_= , then

F(β,β_,β_) =α_β_^+α_β_^+α_β_^β_^=β_^β_^

α_ β_

β



+α_+α_ β_

β



() wheneverβ_=  andβ_= . Lett=β_/β. FromF(β,β_,β_)≥, equation () implies

α≥–αt^–α

/t^

, ∀t= , i.e.,

α_≥max

t=

–αt^–α_

/t^

= –min

t=

α_t^+α_

/t^

= –√ α_α_. Furthermore, ifα_≥, then

F(β,β_,β_)≥α_β_^+α_β_^β_^+α_β_β_β_^=

β_^ β_β_

α α/

α_/ α_ β_^ β_β_

≥,

where the last step is due to

α_ α_/

α_/ α_

_S^₊O,

which is ensured by condition (). Similarly, ifα_∈[–√α_α_, ) (implyingα_=  in this case), then

F(β,β_,β_)

= √

α_β_^+ α_

√α_β_^ 

+

α_– α^_

α

β_^+α_β_^β_^+α_β_^β_^+α_β_β_β_^

≥

α_– α_^

α_

β_^+α_β_^β_^+α_β_β_β_^

=

β_^ β_β_

α_– (α_^/α) α_/

α_/ α_ β_^ β_β_

≥,

where the last step is due to

α_– (α_^/α) α_/

α_/ α_

_S^₊O,

which is ensured by condition () and the factα_– (α^_/α)≥ since –√α_α_≤α_< .

This completes the proof.

Lemma .[, Theorem .] Let f :J→Rand f^Lθbe defined as in().Then f^Lθis second- order differentiable at x∈S if and only if f is second-order differentiable atλ_i(x)∈J for

i= , .Moreover,for u,v∈Rⁿ,if x= ,then f^Lθ

(x)(u,v)

=

⎧⎪

⎨

⎪⎩ f(x)

u^Tv

uv+vu , either u= or v= , f(x)u^Tv

f(x)(vu+uv)+^_f(x)(tanθ–cotθ)(uv+¯u^T_¯vvu) , otherwise.

If x_= ,then f^Lθ

(x)(u,v) =

I

I

,

where

I:=vuξ˜+˜

ux¯^T_v+vx¯^T_u

+av˜ ^T_u+ (η˜–a)¯˜ x^T_vx¯^T_u, I:=

(η˜–a)u˜ x¯^T_v+ ( – d)¯˜ x^T_vx¯^T_u+v˜ u+ (η˜–a)v˜ x¯^T_u

x¯

+d˜

¯

x^T_uv+v^T_ux¯+x¯^T_vu

+a(u˜ v+vu)

with

˜

a=f(λ(x)) –f(λ(x)) λ_(x) –λ_(x) , ξ˜= f(λ(x))

 +cot^θ +f(λ(x))

 +tan^θ,

˜

= – cotθ

 +cot^θf λ_(x)

+ tanθ

 +tan^θf λ_(x)

,

˜

η= cot^θ

 +cot^θf λ_(x)

+ tan^θ

 +tan^θf λ_(x)

, d˜= 

x

cot^θ

 +cot^θf λ_(x)

+ tan^θ

 +tan^θf λ_(x)

–f(λ(x)) –f(λ(x)) λ_(x) –λ_(x)

,

= – cot^θ

 +cot^θf λ_(x)

+ tan^θ

 +tan^θf λ_(x)

.

The characterization ofLθ-convexity is established below, which can be regarded as the extension of some results given in [, –] from the second-order cone setting to the circular cone setting.

Theorem . Suppose that f :J→Ris second-order continuously differentiable.If f is Lθ-convex of order n on S,thentanθ≥,f is convex on J,and for allτ_,τ_∈J withτ_≤τ_,

f(τ)δ(τ,τ)≥ 

(τ–τ_)^δ(τ,τ)^ ()

and

tan^θ δ(τ_,τ_) +

tan^θ– 

δ(τ_,τ_)

f(τ_) – 

(τ–τ_)^δ(τ_,τ_)^

≥–f(τ)

tan^θ– 

δ(τ,τ_)

tan^θ δ(τ,τ_) +

tan^θ– 

δ(τ,τ_)

. ()

Furthermore,if tan^θ δ(τ,τ_) +

tan^θ– 

δ(τ,τ_)

f(τ)≥ 

(τ–τ)^δ(τ,τ_)^ () and

δ(τ,τ_)δ(τ,τ_)^≤

tan^θ δ(τ,τ_) +

tan^θ– 

δ(τ,τ_)

f(τ)f(τ)(τ–τ_)^, () or if

tan^θ δ(τ_,τ_) +

tan^θ– 

δ(τ_,τ_)

f(τ_) < 

(τ–τ_)^δ(τ_,τ_)^ and

δ(τ_,τ_)^δ(τ_,τ_)^

tan^θ–  f(τ_)

≤

tan^θ–  f(τ_)^

tan^θ δ(τ_,τ_) +

tan^θ– 

δ(τ_,τ_)

δ(τ_,τ_)

–

tan^θ δ(τ,τ_) +

tan^θ– 

δ(τ,τ_)

f(τ) – 

(τ–τ)^δ(τ,τ_)^ 

×f(τ)(τ–τ_)^, ()

then f isLθ-convex.Hereδ(τ,τ) :=f(τ) –f(τ) –f(τ)(τ–τ)forτ,τ∈J.

Proof According to [, Theorem .],f isLθ-convex if and only if (f^Lθ)(x)(h,h)∈Lθfor allx∈Sandh∈Rⁿ. We proceed the proof by considering the following three cases.

Case . Forx=  andh= , it follows from Lemma . that f^Lθ

(x)(h,h) =f(x) h^_



.

Hence, (f^Lθ)(x)(h,h)∈Lθif and only iff(x)≥.

Case . Forx_=  andh_= , it follows from Lemma . that f^Lθ

(x)(h,h) =

f(x)h^

f(x)hh+f(x)(tanθ–cotθ)hh

.

Hence, (f^Lθ)(x)(h,h)∈Lθif and only iff(x_)≥ and tanθh^≥h+ (tanθ–cotθ)hh, i.e.,

–tanθ

h^_+h^

≤

h+ (tanθ–cotθ)h

h ≤tanθ

h^_+h^ .

Dividing byh^and lettingt=h/hyields –tanθ

t^+ 

≤t+tanθ–cotθ≤tanθ t^+ 

⇐⇒ max

t∈R –tanθ t^+ 

– t≤tanθ–cotθ≤min

t∈Rtanθ t^+ 

– t

⇐⇒ cotθ–tanθ≤tanθ–cotθ≤tanθ–cotθ

⇐⇒ tanθ≥.

Case . Forx_= , due to the simplification of notation, let us denote μ_:=h–cotθx¯^T_h, μ_:=h+tanθx¯^T_h, μ_:=

h^–

¯ x^T_h



. ()

Then

¯

x^T_h= μ_–μ_

tanθ+cotθ and h=tanθ μ_+cotθ μ_

tanθ+cotθ . ()

Note thatμ_,μ_, andμ_can take any value inR×R×R+by taking a suitable value ofh (because the vectorhhasnvariables). It follows from Lemma . that

f^Lθ (x)(h,h)

=

ξ˜h^_+ ¯˜x^T_hh+a˜h^+ (η˜–a)(¯˜ x^T_h)^

[( – d)(˜ x¯^T_h_)^+ (η˜–a)˜ x¯^T_h_h_]x¯_+ [h˜ ^_+d˜h_^]x¯_+ [ah˜ _+d˜x¯^T_h_]h_

=:

_{ }x¯+_h

,

where

_=ξ˜h^_+ ˜x¯^T_hh+ah˜ ^+ (η˜–a)˜

¯ x^T_h



, _= ( – d)˜

¯ x^T_h_

+ (η˜–a)˜ x¯^T_h_h_+h˜ ^_+d˜h_^, _= 

˜

ah+d¯˜x^T_h

.

Hence, (f^Lθ)(x)(h,h)∈Lθis equivalent to _≥ and ^_tan^θ≥ _x¯+_h^. Note that

_= 

 +cot^θf λ_(x)

h^_– 

¯ x^T_h

hcotθ+

¯ x^T_h



cot^θ

+ 

 +tan^θf λ_(x)

h^_+ 

¯ x^T_h_

h_tanθ+

¯ x^T_h_

tan^θ +a˜

h_^–

¯ x^T_h_

= 

 +cot^θf λ_(x)

μ^_+ 

 +tan^θf λ_(x)

μ^_+aμ˜ ^_. ()

We now claim that_≥ for allh∈Rⁿif and only if f

λ_(x)

≥, f λ_(x)

≥, and a˜≥. () The sufficiency is clear. Let us show the necessity. In particular, choosingh= (–tanθ,x¯) yieldsμ_=  andμ_= . It then follows from_≥ that f(λ(x))≥. If we choose h= (cotθ,x¯), then we havef(λ(x))≥. Finally, choosingh= (,kz) withk∈R,z=  andz^T_x¯_=  gives

_= f(λ(x))

 +cot^θ +f(λ(x))

 +tan^θ +ak˜ ^≥.

Dividing byk^ both sides and taking the limits ask→ ∞, we obtaina˜≥. Sinceλ_i(x) can take an arbitrary value inJ, it is clear that () is equivalent to saying thatf(τ)≥ for allτ ∈J,i.e.,f is convex onJ. Indeed, the conditiona˜≥ is ensured by the fact that

˜

a=^f(λ_λ^{(x))–f(λ(x))}

(x)–λ(x) =f(t)≥ for somet∈(λ(x),λ_(x)).

Now we calculate the values of_and_, respectively.

= – cotθ

 +cot^θf λ(x)

μ^_+ tanθ

 +tan^θf λ(x)

μ^_ +dμ˜ ^_– d˜x¯^T_h_+ah˜ _

¯ x^T_h_

= – cotθ

 +cot^θf λ_(x)

μ^_+ tanθ

 +tan^θf λ_(x)

μ^_+dμ˜ ^_–

¯ x^T_h

_. ()

Meanwhile, it follows from () that = 

˜

atanθ μ_+cotθ μ_

tanθ+cotθ +d˜ μ_–μ_ tanθ+cotθ

= 

tanθ+cotθ

μ(a˜tanθ–d) +˜ μ(˜acotθ+d)˜

. ()

Note that

_x¯+_h^=^_+ _x¯^T_h+^_h^

=^_+ _x¯^T_h+^_ μ^_+

¯ x^T_h



=

+x¯^T_h



+^_μ^_. ()

Putting () and ()-() together, the condition ^_tan^θ ≥ _x¯ +_h^ can be rewritten equivalently as

tan^θ

f(λ(x))

 +cot^θμ^_+f(λ(x))

 +tan^θμ^_+aμ˜ ^_ 

≥

– cotθ

 +cot^θf λ_(x)

μ^_+ tanθ

 +tan^θf λ_(x)

μ^_+dμ˜ ^_ 

+ 

(tanθ+cotθ)^

μ_(˜atanθ–d) +˜ μ_(a˜cotθ+d)˜ 

μ^_,

i.e.,

tan^θ–  f

λ_(x)

μ^_+ (tanθ+cotθ)^

˜

a^tan^θ–d˜^ μ^_ + 

(tanθ+cotθ)

˜

atan^θ+d˜ f

λ_(x)

– (a˜tanθ–d)˜ ^ μ^_μ^_ + 

(tanθ+cotθ)(a˜tanθ–d)f˜ λ_(x)

– (a˜cotθ+d)˜ ^ μ^_μ^_ + 

tan^θ+  f

λ_(x) f

λ_(x) μ^_μ^_

– (a˜tanθ–d)(˜˜ acotθ+d)μ˜ μ_μ^_≥. ()

To apply Lemma ., we need to compute each coefficient in (). By calculation, we have

˜

atanθ–d˜

=f(λ(x)) –f(λ(x))

λ_(x) –λ_(x) tanθ–  x_

cot^θ

 +cot^θf λ_(x)

+ tan^θ

 +tan^θf λ_(x)

+ 

x_

f(λ(x)) –f(λ(x)) λ_(x) –λ_(x)

=f(λ_(x)) –f(λ_(x))

λ(x) –λ(x) tanθ–  x

cotθ tanθ+cotθf

λ_(x)

+ tanθ tanθ+ctanθf

λ_(x)

+ 

x

f(λ(x)) –f(λ(x)) λ(x) –λ(x)

= –tanθ+ctanθ λ_(x) –λ_(x)f

λ_(x)

+ tanθ+ctanθ [λ(x) –λ_(x)]^

f λ_(x)

–f λ_(x)

=(tanθ+cotθ)[f(λ(x)) –f(λ(x)) –f(λ(x))(λ(x) –λ_(x))]

[λ_(x) –λ_(x)]^

= tanθ+cotθ [λ(x) –λ_(x)]^δ

λ_(x),λ_(x) ,

where the third equation follows from the factλ_(x) –λ_(x) = (tanθ+ctanθ)x. Similarly, we have

˜ atanθ+d˜

=(tanθ+cotθ)[f(λ(x)) –f(λ(x)) + (_tanθ+cot^tanθ_θf(λ(x)) +^cot_tanθ+cot^θ–tanθ_θf(λ(x)))(λ(x) –λ_(x))]

[λ(x) –λ_(x)]^

= tanθ+cotθ [λ(x) –λ_(x)]^

tan^θ tan^+δ

λ(x),λ(x)

+tan^θ–  tan^θ+ δ

λ(x),λ(x) ,

˜

acotθ+d˜=(tanθ+cotθ)[f(λ(x)) –f(λ(x)) –f(λ(x))(λ(x) –λ(x))]

[λ(x) –λ(x)]^

= tanθ+cotθ [λ(x) –λ(x)]^δ

λ_(x),λ_(x) ,

˜

atan^θ+d˜

=(tanθ+cotθ)[f(λ(x)) –f(λ(x)) – [tan^θf(λ(x)) + ( –tan^θ)f(λ(x))](λ(x) –λ(x))]

[λ(x) –λ(x)]^

= tanθ+cotθ [λ(x) –λ(x)]^

tan^θ δ

λ(x),λ(x) +

tan^θ–  δ

λ(x),λ(x) .

Corresponding each coefficient in () to (), we know

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

α_= (tan^θ– )f(λ(x))^, α_= (tanθ+cotθ)^

[λ(x)–λ(x)]^δ(λ(x),λ_(x))[_tan^tanθ+^θδ(λ(x),λ_(x)) +^tan_tan^^θ–θ+δ(λ(x),λ_(x))], α_= (tanθ+cotθ)^

[λ(x)–λ(x)]^{[tan^θ δ(λ(x),λ_(x)) + (tan^θ– )δ(λ(x),λ_(x))]f(λ(x)), – ^δ(λ_[λ^{(x),λ(x))}

(x)–λ(x)]^}, α_= (tanθ+cotθ)^

[λ(x)–λ(x)]^[δ(λ(x),λ_(x))f(λ(x)) – ^δ(λ_[λ^{(x),λ(x))}

(x)–λ(x)]^], α_= (tan^θ+ )f(λ(x))f(λ(x)),

α_= –(tanθ+cotθ)^

[λ(x)–λ(x)]^δ(λ_(x),λ_(x))δ(λ_(x),λ_(x)).

In view of Lemma ., the conditionα_≥ meanstanθ≥,α_,α_≥ is ensured by the convexity off(see ()),α≥ corresponds to (), andα≥–√

ααcorresponds to ().

In addition, condition () takes the special form () and (), respectively.

Theorem . Suppose that f :J→Ris second-order continuously differentiable.Then f isLθ-convex of orderon S if and only iftanθ≥and f is convex on J.

Proof The necessity is clear from Theorem .. For sufficiency, note that in ()μ_=  sincex¯=± in this case. Hence, () takes the form of

tan^θ–  f

λ(x)

μ^_+ 

tan^θ+  f

λ(x) f

λ(x)

μ^_μ^_≥ for allμ_andμ_, which is equivalent to verifying

tanθ≥ and f λ_(x)

f λ_(x)

≥.

This is ensued by the conditions thattanθ≥ andf is convex on J. Thus, the proof is

complete.

If, in particular,θ = ^◦, then () and () reduce to [, () in Proposition .]; () re- duces to [, () in Proposition .]. In addition, due to (), () holds automatically in this case. The above results indicate that theLθ-convexity is dependent on the properties off and the angleθ together.

3 Inequalities associated with circular cone

In this section, we establish some inequalities associated with circular cone, which we be- lieve will be useful for further analyzing the properties off^Lθand proving the convergence of interior point methods for optimization problems involved in circular cones.

In [], the author establishes the following results in the framework of second-order cone. More specifically, forxL◦  andyL◦ , then

(a) det(e+x)^/≥ +det(x)^/, (b) det(x+y)≥det(x) +det(y),

(c) det(αx+ ( –α)y)≥α^det(x) + ( –α)^det(y),∀α∈[, ], (d) det(e+x+y)≤det(e+x)det(e+y),

(e) IfxL◦ yL◦ , thendet(x)≥det(y),tr(x)≥tr(y), andλ_i(x)≥λ_i(y)fori= , , (f ) tr(x+y) =tr(x) +tr(y)anddet(γx) =γ^det(x)for allγ∈R.

In the following, we show that, in the framework of circular cone, the above inequalities can be classified into three categories. The first class holds independent of the angle,e.g., (a); the second class holds dependent on the angle, e.g., (b)-(e); the third class fails no matter what value of the angle is chosen,e.g., (f ).

Theorem . Let x= (x,x)∈R×R^n–possess spectral factorization associated with cir- cular cone given as in().Then

(a) [det(e+x)]^/≥ +det(x)^/for allx∈Lθ; (b) IfxLθ y,thenλ_(x)≥λ_(y).

Proof (a) Note thatdet(x)≥ anddet(e+x)≥ sincex,x+e∈Lθ. Therefore, det(e+x)/

≥ +det(x)^/

⇐⇒ det(e+x)≥ + det(x)^/+det(x)

⇐⇒ λ_(e+x)λ(e+x)≥ + 

λ_(x)λ(x) +λ_(x)λ(x)

⇐⇒

x+  –xcotθ

x+  +xtanθ

≥ + 

λ_(x)λ(x) +λ_(x)λ(x)

⇐⇒

λ_(x) + 

λ_(x) + 

≥ + 

λ_(x)λ(x) +λ_(x)λ(x)

⇐⇒ λ(x)λ(x) +λ(x) +λ(x) + ≥ + 

λ(x)λ(x) +λ(x)λ(x)

⇐⇒ λ_(x) +λ_(x)≥

λ_(x)λ_(x)

⇐⇒ λ_(x) +λ_(x)

 ≥

λ_(x)λ(x).

Hence, to prove the desired result, it suffices to show that λ_(x) +λ_(x)

 ≥

λ_(x)λ_(x),

which is clearly true by the arithmetic mean-geometric mean (AM-GM) inequality.

(b) Sincex–y∈Lθ, we know x–y≥ x–ycotθ≥

x–y cotθ,

i.e.,λ_(x) =x–xcotθ≥y–ycotθ=λ_(y).

Theorem . Let x= (x,x)∈R×R^n–possess spectral factorization associated with cir- cular cone given as in().Then the following hold.

(a) For allx,y∈Lθ,

det(x+y)≥det(x) +det(y) +

x^+y^

csc^θ– x^_+y^_

sec^θ. In particular,whenθ∈(, ^◦],we have

det(x+y)≥det(x) +det(y). ()