This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.

1. International Journal of Innovation and Scientific Research
ISSN 2351-8014 Vol. 4 No. 1 Jul. 2014, pp. 42-46
© 2014 Innovative Space of Scientific Research Journals
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Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold
JBILOU Asma
Pôle énergies renouvelables,
Université Internationale de Rabat (UIR),
Laboratoire des Energies Renouvelables et Matériaux Avancés (ERMA),
Parc Technopolis Rabat-Shore, campus de l’UIR, Rocade Rabat-Salé,
11100, Rabat-Sala El Jadida, Morocco
Copyright © 2014 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT: We prove in this article using some convex analysis results of A. S. Lewis, the log-concavity of spectral elementary
symmetric functions on the space of Hermitian matrices, and the convexity of the set of k-admissible functions on compact
Kähler manifolds.
KEYWORDS: Spectral functions, symmetric functions, log-concavity, convexity, admissible functions, hessian equations, Kähler
manifolds.
1 INTRODUCTION AND STATEMENT OF RESULTS
All manifolds considered in this article are connected.
Let (, , g, ω) be a compact connected Kähler manifold of complex dimension ≥ 1. Fix an integer 1 ≤ k ≤ m. Let
∶ M → ℝ be a smooth function and let us consider the (1,1)-form ῶ = ω + i ∂∂
φ and the associated 2-tensor defined
by (, ) = ῶ(, ). Consider the sesquilinear forms ℎ and ℎ
on !,# defined by ℎ($, %) = g(U,V(
) and ℎ
(U, V) = (U,V(
).
We denote ℎby )(*) the eigenvalues of with respect to the hermitian form ℎ. By definition, these are the eigenvalues of
the unique endomorphism + of !,# satisfying: (U, V) = h(U, AV) ∀U, V ∈ !,#.
ℎCalculations infer that the endomorphism + writes:
+ ∶ !,# → !,#
3$123=34̅14̅$123
$121 → +1
+ is a self-adjoint/hermitian endomorphism of the hermitian space (!,#, ℎ), therefore )(*) ∈ ℝ6.
Let us consider the following cone: Γ8 = 9) ∈ ℝ6, ∀1 ≤ : ≤ ; 3()) 0? , where 3 denotes the :-th elementary
symmetric function.
Definition. is said to be ;-admissible if and only if )(*) ∈ Γ8.
In a note in the Comptes Rendus de l’Académie des Sciences de Paris published online in December 2009 [1], we solve the
equations ῶ8˄ A6*8 = BC
DEF
G A6 (H8), when the holomorphic bisectional curvature of is nonnegative. In this proof
performed by the continuity method, two results following from convex analysis techniques were needed, namely the
Corollaries 1.5 and 1.6.
Let us now introduce some convex analysis notations. Let I6(ℂ) be the space of complex Hermitian matrices of order .
We recall that for any two matrices K and L of I6(ℂ), ) ∈ ℂ is called a K-eigenvalue of L if there exists M ≠ 0 in ℂ6 such
Corresponding Author: JBILOU Asma (asma.jbilou@uir.ac.ma) 42

2. JBILOU Asma
that LM = )KM, M is then called a K-eigenvector of L. Let K ∈ I6(ℂ) be a fixed positive definite matrix. Let us recall the
following basic result :
Lemma 1.2. Let L ∈ I6(ℂ), then:
1. The spectrum of K*L (i.e. the K-spectrum of L) is entirely real.
2. The greatest eigenvalue of K*L (i.e. the greatest K-eigenvalue of L) equals supRS#
TUR,RV
TWR,RV , where . , . denotes the
standard Hermitian product of ℂ6.
3. K*L is diagonalizable.
Since the spectrum of K*L is the spectrum of the Hermitian matrix K*Y
ZLK*Y
Z, the proof is an easy adaptation of the
standard one for symmetric matrices.
For a given Hermitian matrix L, we denote by )W(C) the eigenvalues of L with respect to K. In this article, we prove the
following four results using the Theorem 2.3 and the Corollary 2.4 of Lewis [2] (see Theorem 2.1 of this article):
Theorem 1.3. For each ; ∈ {1, . . . , m}, the function:
^8
W: I6(ℂ) → ℝ ∪ {+∞}, L → ^8
W(L) = b− ln 8D)W(L)G fg L ∈ )W
*(Γ8)
+∞ otherwise,
where )W
*(Γ8) = {L ∈ I6(ℂ), )W(C) ∈ Γ8}, is convex.
Theorem 1.4. If Γ is a (non empty) symmetric convex closed set of ℝ6, then )W
*(Γ): = {L ∈ I6(ℂ), )W(C) ∈ Γ},
is a convex closed set of I6(ℂ). In particular, )W
*(Γ8 ( ) is a convex closed set of I6(ℂ).
By the Theorem 1.3, and since )W
*(Γ8) is convex (Theorem 1.4), we deduce that:
Corollary 1.5. The function:
−^8
W: )W
*(Γ8) → ℝ, L → ^8
W(L) = ln 8D)W(L)G is concave.
The method used here to prove the Corollary 1.5, gives for K = m a different approach from the proof of [3] and the
elementary proof of [4, p. 51] and [5].
As an immediate consequence of Theorem 1.4, we get the following important result, that allows to notably simplify the
proof of ∂uniqueness of the solution of the equation (H8) in comparison with [5]:
Corollary 1.6. For a compact connected Kähler manifold (, , g, ω), the set of ;-admissible functions:
A8 = 9 ∈ Ln(, ℝ), )oDω + i ∂φG ∈ Γ8? is convex.
2 SOME CONVEX ANALYSIS
The space I6(ℂ) has a structure of Euclidean space thanks to the following scalar product ≪ +, K ≫= tr( A(
r B) =
tr(AB), called the Schur product. Let us denote by t#(ℝ6) the set of functions u: ℝ6 → ℝ ∪ {+∞} that are convex, lower
semicontinuous on ℝ6, and finite in at least one point. Given u ∈ t#(ℝ6) symmetric and K ∈ I6(ℂ) positive definite, we
define:
W: I6(ℂ) → ℝ ∪ {+∞}, vw L → %R
%R
W(L): = u()W,(L), … , )W,6(L))
W
where )W,(L) ≥ )W,n(L) ≥ … ≥ )W,6(L)denote the B-eigenvalues of C repeated with their multiplicity. Such functions %R
are called functions of B-eigenvalues or B-spectral functions. Our first aim is to determine the conjugation for such a
function %R
W using the conjugate function of u. Let us remind that the conjugation or the Legendre–Fenchel transform of u is
the function u∗: ℝ6 → ℝ ∪ {+∞} defined by:
∀s∈ ℝ6, u∗(z) = sup{∈ℝE{‹z, M› − u(M)}
where ‹. , . › denotes the standard scalar product on ℝ6.
Theorem 2.1 (A. S. Lewis [2], Conjugation of spectral functions). Let u ∈ t#(ℝ6) be symmetric, then:
1. The conjugate u∗ (∈ t#(ℝ6)) is also symmetric.
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3. Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold
~ (defined as above) belong to t#DI6(ℂ)G with %R∗
~ and %R∗
2. The functions of eigenvalues %R
~ = (%R
~)∗, so that in particular the
~ is convex and lower semicontinuous.
function of eigenvalues %R
Proof. See the Theorem 2.3 and the Corollary 2.4 of Lewis [2]. ∎
A similar theorem is proved in the case of symmetric matrices in [6] and [7] (you can see also [8] and [9] for some details).
Corollary 2.2 (Conjugation of K-spectral functions). Let u ∈ t#(ℝ6) be symmetric, then:
1. The conjugate u∗ (∈ t#(ℝ6)) is also symmetric.
2. The functions of K-eigenvalues %R
W€Y = (%R
W (defined as above) belong to t#DI6(ℂ)G with %R∗
W and %R∗
W)∗, so that in
W is convex and lower semicontinuous.
particular the function of K-eigenvalues %R
3 PROOF OF THEOREMS 1.3 AND 1.4.
3.1 PROOF OF THEOREM 1.3.
The proof of Theorem 1.3 is a direct application of the Corollary 2.2 to the function:
u : ℝ6 → ℝ ∪ {+∞}, M = (M,… , M6) → u(x) = b− ln 8(M, … , M6) fg M ∈ t8
+∞ ‚ƒℎ„…†fz„
(3.1)
Our function u is symmetric and belongs to t#(ℝ6), indeed:
(i) It is clearly symmetric. It is finite in a least one point of ℝ6 because t8 is non empty. And it is convex, because the
function (8)
Y
F ∶ t8 → ℝ is concave [3, p.269].
(ii) It is lower semicontinuous. Indeed, let ‡ ∈ ℝ, and consider the set:
{M ∈ ℝ6/ +∞ ≥ u(M) ‡} = {M ∈ t8 / u(M) ‡} ∪ {M ∉ t8/u(M) ‡}
= {M ∈ t8 /− Š‹ 8(M) ‡} ∪ (ℝ6 ∖ t8) (3.2)
By continuity, {M ∈ t8 / − Š‹ 8(M) ‡} is an open set of t8, it is then an open of ℝ6 since t8 is an open of ℝ6.
Furthermore, the cone t8 is also a closed set of ℝ6 (as a connected component), consequently ℝ6 ∖ t8 is an open set of ℝ6.
Therefore, {M ∈ ℝ6/ +∞ ≥ u(M) ‡} is an open set of ℝ6 too. This is valid for all ‡ ∈ ℝ, so that u is lower
semicontinuous.
Therefore, we deduce by the Corollary 2.2 that the K-spectral function %R
W is convex, which proves the theorem.
W=^8
Let us remark that the same technique allows to prove for example that the functions
%(L) := the greatest K-eigenvalue of L and
%(L) := the sum of the z greatest K-eigenvalues of L
with z ∈ {1, . . . , }, (3.3)
are convex on I6(ℂ).
3.2 PROOF OF THEOREM 1.4.
The proof of Theorem 1.4 goes by considering the indicatrix function g# ∶= mŽ of the set t, namely:
g# ∶= mŽ ∶ ℝ6 → ℝ ∪ {+∞}, M = (M,… , M6) → mŽ (M) =  0 fg M ∈ t
+∞ ‚ƒℎ„…†fz„ (3.4)
From the assumptions made on t, g# lies in t#(ℝ6) and is symmetric, indeed:
(i) This function is clearly finite in at least one point since t is non empty.
(ii) The inequality mŽ(ƒM + (1 − ƒ)w) ≤ ƒ mŽ(M) + D1 – ƒG mŽ(w) is valid for all M, w ∈ ℝ6 and all ƒ ∈ [0, 1]. Indeed, if
M, w ∈ t then ƒM + (1 − ƒ)w ∈ t by convexity of tand the two sides of the convexity inequality equal 0 in this
case. Furthermore, if M or w does not belong to t then the right side of the inequality equals +∞ and the inequality
is then satisfied in this case too, which proves that mŽ is convex.
(iii) mŽ is lower semicontinuous. Indeed, let “ ∈ ℝ6 : If “ ≥ 0 then {M ∈ ℝ6 /+∞ ≥ mŽ(M) “} = ℝ6 ∖ t is an open
set since t is closed. Besides, if a 0, {M ∈ ℝ6 /+∞ ≥ mŽ(M) “} = ℝ6 is an open set too.
W lies in t#(ℝ6); in particular it is, convex lower semicontinuous.
So Corollary 2.2 implies that the function of K-eigenvalues %~”
But this function is given by:
%~”
W (L) = b0 fg L ∈ )W
W : I6(ℂ) → ℝ ∪ {+∞}, L → %~”
*(Γ)
+∞ ‚ƒℎ„…†fz„
(3.5)
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4. JBILOU Asma
In other words, it coincides with m•–
€Y(—), the indicatrix function of )W
*(Γ). So the latter must itself be convex lower
semicontinuous. As a consequence, )W
*(Γ)is a convex closed (non empty) set of I6(ℂ), indeed:
(i) )W
*(Γ) is convex because if L, ˜ ∈ )W
*(Γ) and ƒ ∈ [0,1], we have by convexity of m•–
€Y(—),
m•–
€Y(—)(ƒL + (1 − ƒ)˜) ≤ ƒm•–
€Y(—)(L) + (1 − ƒ) m•–
€Y(—)(˜) (3.6)
But m•–
€Y(—)(L) = m•–
€Y(—)(˜) = 0 then necessarily m•–
€Y(—)(ƒL + (1 − ƒ)˜) = 0 and ƒL + (1 − ƒ)˜ ∈ )W
*(Γ).
(ii) The set )W
*(Γ) is closed because  ∈ I6(ℂ)/ +∞ ≥ m•–
€Y(—)() 0™ = I6(ℂ) ∖ )W
*(Γ) is an open set since
m•–
€Y(—) is lower semicontinuous.
4 SIMPLIFICATION OF THE PROOF OF UNIQUENESS OF THE SOLUTION OF (š›).
The Corollary 1.6 allows to notably simplify the proof of uniqueness of the solution of the equation (H8) in comparison
with [5].
Let # and be two smooth ;-admissible solutions of the equation (H8 ) such that ∫ # A6
 =∫ A6
 = 0. For all
ƒ ∈ [0, 1], let us consider the function ž = ƒ + (1 − ƒ) # = # + ƒ with = − #. Let Ÿ ∈ , and let us
denote ℎ3 + 34̅(8
Ÿ)214̅ž (Ÿ)£G. We have ℎ8
(ƒ) = g8D¡¢1
(1) − ℎ8
¤(ƒ)¥ƒ = 0
# . But:
(0) = 0 which is equivalent to ∫ ℎ8
6
6
¤(ƒ) = ¦ §¦
ℎ8
2g8
2K1
4
4¨
3 + 34̅(Ÿ)214̅ž(Ÿ)£G 4©̅(Ÿ)ª
D¡¢1
1,3¨
21©̅(Ÿ)
ž (Ÿ) = Σ ­®F
Let us denote «13
3 + 34̅(Ÿ)214̅ž(Ÿ)£G 4©̅(Ÿ). Therefore we obtain:
¨ °
D¡¢1
­W¯
64
ž (Ÿ)
# ¥ƒ
±(Ÿ) ≔ Σ “13(Ÿ) 61
,3¨ 21©̅(Ÿ) = 0 with “13 (Ÿ) = ∫ «13
We show easily that the matrix ¡“13(Ÿ)£³1,3³6 is hermitian [4, p. 53]. By the Corollary 1.6, we know that for all ƒ ∈ [0, 1] and
all points ∈ , )D*´µ ¶ G( ) ∈ t8, namely that the functions (ž)ž∈[#,] are ;-admissible.
We check then easily that the hermitian matrix ¡“13( )£³1,3³6 is positive definite for all ∈ [4, p. 54]. Consequently,
the operator ± is elliptic on . But the map is L· and satisfies ± = 0, then by the Hopf maximum principle [10], we
deduce that is constant on . Besides ∫ A6 = 0  , therefore we deduce that ≡ 0 on namely that ≡ # on ,
which achieves the proof of uniqueness.
ACKNOWLEDGMENT
The present results are an auxiliary, but independent, part of my PhD dissertation [4].
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5. Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold
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