TAIWANESE JOURNAL OF MATHEMATICS
Vol. 20, No. 4, pp. 909–919, August 2016
DOI: 10.11650/tjm.20.2016.6323
This paper is available online at http://journal.tms.org.tw
The Dual Log-Brunn-Minkowski Inequalities
Wei Wang* and Lijuan Liu
Abstract. In this article, we establish the dual log-Brunn-Minkowski inequality and
the dual log-Minkowski inequality. Moreover, the equivalence between the dual logBrunn-Minkowski inequality and the dual log-Minkowski inequality is demonstrated.
1. Introduction
As a cornerstone of the Brunn-Minkowski theory, the classical Brunn-Minkowski inequality
provides a beautiful and powerful apparatus for conquering all sorts of geometrical problems involving metric quantities such as volumes, surface area, and mean width (see [3]).
The classical Brunn-Minkowski inequality states that for convex bodies K and L in
Euclidean n-space, Rn , the volume of the bodies and their Minkowski sum K + L =
{x + y : x ∈ K, y ∈ L}, are related by
1
1
1
V (K + L) n ≥ V (K) n + V (L) n ,
with equality if and only if K and L are homothetic.
The Brunn-Minkowski inequality exposes the crucial log-concavity property of the
volume functional because the Brunn-Minkowski inequality has an equivalent formulations:
for 0 ≤ λ ≤ 1,
(1.1)
V ((1 − λ)K + λL) ≥ V (K)1−λ V (L)λ ,
and for 0 < λ < 1, there is equality if and only if K and L are translates.
In the early 1960s, Fiery [2] defined the Minkowski-Fiery Lp -combinations (or simply
Lp -Minkowski combinations) of convex bodies. If K and L be two convex bodies that
Received May 24, 2015; Accepted September 8, 2015.
Communicated by Duy-Minh Nhieu.
2010 Mathematics Subject Classification. 52A40, 53A15.
Key words and phrases. Star body, Radial function, Log radial sum, Dual Brunn-Minkowski inequality,
Dual Minkowski inequality.
A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (11C0542)
and the National Natural Science Foundations of China (NO. 11371239 and 11471206)
*Corresponding author.
909
910
Wei Wang and Lijuan Liu
contain the origin in their interiors, p ≥ 1, and 0 ≤ λ ≤ 1, then the Lp -Minkowski
combinations, (1 − λ) · K +p λ · L, is defined by
o
\ n
1
(1.2) (1 − λ) · K +p λ · L =
x ∈ Rn : x · u ≤ ((1 − λ)hK (u)p + (λ)hL (u)p ) p ,
u∈S n−1
where x · u denotes the standard inner product of x and u in Rn , and hK is the support
function of K.
Fiery also established the Lp -Brunn-Minkowski inequality. If p > 1, then
V ((1 − λ)K +p λL) ≥ V (K)1−λ V (L)λ ,
with equality for 0 < λ < 1 if and only if K = L.
1
Note that definition (1.2) makes sense for all p > 0. The function ((1 − λ)hpK + λhpL ) p
1
is convex if p ≥ 1. Whereas the function ((1 − λ)hpK + λhpL ) p is not convex if 0 < p < 1.
1
λ
+
The limit of ((1 − λ)hpK + λhpL ) p is h1−λ
K hL , as p → 0 .
Recently, Böröczky et al. [1] defined the log Minkowski combination of convex bodies.
Let K and L be two convex bodies that contain the origin in their interiors, and 0 ≤ λ ≤ 1,
then the log Minkowski combination, (1 − λ) · K +0 λ · L, is defined by
o
\ n
(1.3)
(1 − λ) · K +0 λ · L =
x ∈ Rn : x · u ≤ hK (u)1−λ hL (u)λ .
u∈S n−1
It is obviously that the convex body (1 − λ) · K +0 λ · L is the Wolff shape of the function
hK (u)1−λ hL (u)λ .
Böröczky et al. [1] established the log-Brunn-Minkowski inequality and log-Minkowski
inequality for origin-symmetric convex bodies in the plane as follows.
Theorem 1.1. If K and L are two origin-symmetric convex bodies in R2 , then for 0 ≤
λ ≤ 1,
(1.4)
V ((1 − λ) · K +0 λ · L) ≥ V (K)1−λ V (L)λ .
When 0 < λ < 1, equality in the inequality holds if and only if K and L are dilates or K
and L are parallelograms with parallel sides.
Theorem 1.2. If K and L are two origin-symmetric convex bodies in R2 , then
Z
hL (u)
1
V (L)
(1.5)
log
dV K (u) ≥ log
,
hK (u)
2
V (K)
S1
with equality if and only if, K and L are dilates or K and L are parallelograms with parallel
sides. Here V K is the cone-volume probability measure of K.
The Dual Log-Brunn-Minkowski Inequalities
911
Unfortunately, the log-Brunn-Minkowski inequality (1.4) cannot hold for all convex
bodies (e.g., an origin-centered cube and one of its translates). From the arithmeticgeometric mean inequality, it is easily seen that the log-Brunn-Minkowski inequality (1.4)
is stronger than the Brunn-Minkowski inequality (1.1) for origin-symmetric convex bodies.
For n ≥ 3, Böröczky et al. [1] conjectured that there exists the log-Brunn-Minkowski
inequality and log-Minkowski inequality for origin-symmetric convex bodies in Rn , and
showed that these two inequalities are equivalent.
Conjecture 1.3. If K and L are two origin-symmetric convex bodies in Rn (n ≥ 3), then
for 0 ≤ λ ≤ 1,
V ((1 − λ) · K +0 λ · L) ≥ V (K)1−λ V (L)λ .
(1.6)
Conjecture 1.4. If K and L are two origin-symmetric convex bodies in Rn (n ≥ 3), then
Z
1
V (L)
hL (u)
dV K (u) ≥ log
.
(1.7)
log
h
(u)
n
V
(K)
n−1
K
S
The dual Brunn-Minkowski theory, was introduced by Lutwak [7] in the 1970s, helped
achieving a major breakthrough in solving the Busemann-Petty problem in 1990s. In
contrast to the Brunn-Minkowski theory, in the dual theory, convex bodies are replaced
by star bodies, Minkowski sum replaced by radial sum, and mixed volumes are replaced
by dual mixed volumes.
Let K and L be two star bodies about the origin in Rn . For p 6= 0 and 0 ≤ λ ≤ 1,
e p λ · L by
Gardner [4] defined the Lp radial sum (1 − λ) · K +
p
p
p
ρ(1−λ)·K +
e p λ·L (u) = (1 − λ)ρK (u) + λρL (u) ,
∀ u ∈ S n−1 .
Note that
log(1 − λ)ρpK + λρpL
p→0
p
p
(1 − λ)ρK log ρK + λρpL log ρL
= lim
p→0
(1 − λ)ρpK + λρpL
lim log ρ(1−λ)·K +
e p λ·L = lim
p→0
= (1 − λ) log ρK + λ log ρL .
Let K and L be two star bodies in Rn and 0 ≤ λ ≤ 1, then the log radial sum,
e 0 λ · L, is defined by
(1 − λ) · K +
(1.8)
1−λ
ρ(1−λ)·K +
ρL (u)λ ,
e 0 λ·L (u) = ρK (u)
∀ u ∈ S n−1 .
e 0 λ · L = K. If λ = 1, then (1 − λ) · K +
e 0 λ · L = L.
In particular, if λ = 0, then (1 − λ) · K +
The main purpose of this paper is to establish the dual forms of the log-BrunnMinkowski inequality (1.6) and the log-Minkowski inequality (1.7) as follows.
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Wei Wang and Lijuan Liu
Theorem 1.5. If K and L are two star bodies in Rn , then for 0 ≤ λ ≤ 1,
(1.9)
e 0 λ · L) ≤ V (K)1−λ V (L)λ .
V ((1 − λ) · K +
When 0 < λ < 1, equality in the inequality holds if and only if K and L are dilates.
Theorem 1.6. If K and L are two star bodies in Rn , then
Z
ρL (u)
1
V (L)
(1.10)
log
dVeK (u) ≤ log
,
ρ
(u)
n
V
(K)
n−1
K
S
with equality if and only if K and L are dilates. Here VeK is the dual cone-volume probability
measure of K (see Section 3 for a precise definition).
2. Notation and background material
For general reference for the theory of convex (star) bodies the reader may wish to consult
the books of Gardner [4], Gruber [5], and Schneider [9].
The unit ball and its surface in Rn are denoted by B and S n−1 , respectively. We write
V (K) for the volume of the compact set K in Rn . The radial function ρK : S n−1 → [0, ∞)
of a compact star-shaped about the origin, K ∈ Rn , is defined, for u ∈ S n−1 , by
(2.1)
ρK (u) = max {λ ≥ 0 : λu ∈ K} .
If ρK (·) is positive and continuous, then K is called a star body about the origin. The
set of star bodies about the origin in Rn is denoted by S n . Obviously, for K, L ∈ S n ,
(2.2)
K ⊆ L ⇐⇒ ρK (u) ≤ ρL (u),
∀ u ∈ S n−1 .
If ρK (u)/ρL (u) is independent of u ∈ S n−1 , then we say star bodies K and L are dilates.
If s > 0, we have
(2.3)
ρsK (u) = sρK (u),
for all u ∈ S n−1 .
If φ ∈ GL(n), we have
(2.4)
ρφK (u) = ρK (φ−1 u),
for all u ∈ S n−1 .
The radial Hausdorff metric between the star bodies K and L is
e
δ(K,
L) = max |ρK (u) − ρL (u)| .
u∈S n−1
A sequence {Ki } of star bodies is said to be convergent to K if
e i , K) → 0,
δ(K
as i → ∞.
The Dual Log-Brunn-Minkowski Inequalities
913
Therefore, a sequence of star bodies Ki converges to K if and only if the sequence of radial
function ρ(Ki , ·) converges uniformly to ρ(K, ·).
Let K and L be are two star bodies about the origin in Rn and 0 ≤ λ ≤ 1. The radial
e
sum (1 − λ)K +λL
was defined by (see [7])
(2.5)
ρ(1−λ)K +λL
(u) = (1 − λ)ρK (u) + λρL (u),
e
∀ u ∈ S n−1 .
fi (K) has the following integral representation (see [8]):
The dual quermassintegral W
Z
1
f
(2.6)
Wi (K) =
ρK (u)n−i dS(u),
n S n−1
f0 (K) = V (K). The dual
where S is the Lebesgue measure on S n−1 . In particular, W
fi (K, L) has the following integral representation (see [8]):
mixed quermassintegral W
Z
1
f
ρK (u)n−i−1 ρL (u) dS(u).
(2.7)
Wi (K, L) =
n S n−1
By using the Minkowski’s integral inequality, we can obtain the dual Minkowski inequality
for dual mixed quermassintegrals: If K, L ∈ S0n , and 0 ≤ i < n − 1, then
(2.8)
fi (L),
fi (K, L)n−i ≤ W
fi (K)n−i−1 W
W
equality holds if and only if K and L are dilates.
Suppose that µ is a probability measure on a space X and g : X → I ⊂ R is a µintergrable function, where I is a possibly infinite interval. Jessen’s inequality states that
if φ : X → I ⊂ R is a concave function, then
Z
Z
(2.9)
φ(g(x)) dµ(x) ≤ φ
g(x) dµ(x) .
X
X
If φ is strictly concave, equality holds if and only if g(x) is a constant for µ-almost all
x ∈ X (see [6]).
3. Main results
e 0λ · L ∈ S n.
Lemma 3.1. Let K, L ∈ S n and 0 ≤ λ ≤ 1, then (1 − λ) · K +
Proof. Since ρK (·) and ρL (·) are positive and continuous on S n−1 , the function ρK (·)1−λ ·
ρL (·)λ is positive and continuous on S n−1 .
Lemma 3.2. Let K, L ∈ S n and 0 ≤ λ ≤ 1. Then for A ∈ GL(n),
e 0 λ · L) = (1 − λ) · AK +
e 0 λ · AL.
A((1 − λ) · K +
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Wei Wang and Lijuan Liu
Proof. For u ∈ S n−1 , by (1.8) and (2.4), we have
1−λ
ρ(1−λ)·AK +
ρAL (u)λ
e 0 λ·AL (u) = ρAK (u)
= ρK (A−1 u)1−λ ρL (A−1 u)λ
−1
= ρ(1−λ)·K +
e 0 λ·L (A u)
= ρA((1−λ)·K +
e 0 λ·L) (u).
Lemma 3.3. Let Ki , Li ∈ S n and 0 ≤ λ ≤ 1. If Ki → K ∈ S n , Li → L ∈ S n , as i → ∞,
then
e 0 λ · Li → (1 − λ) · K +
e 0 λ · L, as i → ∞.
(1 − λ) · Ki +
Proof. For u ∈ S n−1 , by the continuity of the power function, we have
lim ρ
(u)
e
i→∞ (1−λ)·Ki +0 λ·Li
= lim ρKi (u)1−λ ρLi (u)λ
i→∞
= ρK (u)1−λ ρL (u)λ
= ρ(1−λ)·K +
e 0 λ·L (u).
Lemma 3.4. Let K, L ∈ S n and 0 ≤ λi ≤ 1. If λi → λ ∈ [0, 1], as i → ∞, then
e 0 λi · L → (1 − λ) · K +
e 0 λ · L,
(1 − λi ) · K +
as i → ∞.
Proof. For u ∈ S n−1 , by the continuity of the exponential function, we have
ρL (u) λi
lim ρ
(u) = lim ρK (u)
e
i→∞ (1−λi )·K +0 λi ·L
i→∞
ρK (u)
ρL (u) λ
= ρK (u)
ρK (u)
= ρ(1−λ)·K +
e 0 λ·L (u).
Lemma 3.5. Let K, L ∈ S n and 0 ≤ λ ≤ 1, then
e 0 λ · L ⊆ (1 − λ) · K +λ
e · L,
(1 − λ) · K +
with equality if and only if K = L.
Proof. For u ∈ S n−1 , by the arithmetic-geometric mean inequality and (2.5), we have
1−λ
ρ(1−λ)·K +
ρL (u)λ
e 0 λ·L (u) = ρK (u)
(3.1)
≤ (1 − λ)ρK (u) + λρL (u)
= ρ(1−λ)·K +λ·L
(u).
e
From the equality conditions of the arithmetic-geometric mean inequality, equality in
inequality (3.1) holds if and only if K = L. Combining (3.1) and (2.2), we obtain the
desired result.
The Dual Log-Brunn-Minkowski Inequalities
915
In fact, we will prove the following dual log-Brunn-Minkowski inequality which is more
general than Theorem 1.5.
Theorem 3.6. If K and L are two star bodies in Rn and 0 ≤ λ ≤ 1, then for 0 ≤ i < n,
(3.2)
fi ((1 − λ) · K +
fi (K)1−λ W
fi (L)λ .
e 0 λ · L) ≤ W
W
When 0 < λ < 1, equality in the inequality holds if and only if K and L are dilates.
Proof. By (2.6) and Hölder’s inequality, we obtain that
Z
fi ((1 − λ) · K +
e 0 λ · L) =
W
n−1
ZS
=
ρ(1−λ)·K +λ·L
(u)n−i dS(u)
e
ρK (u)1−λ ρL (u)λ
n−i
dS(u)
S n−1
(3.3)
Z
≤
=
n−i
1−λ Z
dS(u)
ρK (u)
S n−1
fi (K)1−λ W
fi (L)λ .
W
ρL (u)
n−i
λ
dS(u)
S n−1
When 0 < λ < 1, by the equality conditions of Hölder’s inequality, equality in (3.3)
holds if and only if K and L are dilates.
For K ∈ S n , we write the measure Vei,K (·) =
1
fi (K)
nW
(3.4)
Z
S n−1
ρn−i
K (·) dS(·)
fi (K) .
nW
Since
n−i
(u) dS(u) = 1,
ρK
we say that the measure Vei,K (·) is a dual mixed cone-volume probability measure of K on
S n−1 . If i = 0, the measure Ve0,K (·) will be denoted by the dual cone-volume probability
measure, and it will be written simply as VeK (·).
Theorem 3.7. If K and L are two star bodies in Rn , and 0 ≤ i < n, then
Z
(3.5)
log
S n−1
ρL (u)
ρK (u)
dVei,K ≤
fi (L)
1
W
log
,
fi (K)
n−i
W
with equality if and only if K and L are dilates.
Proof. By (2.9), (2.8), and the fact that the logarithmic function log(·) is concave and
916
Wei Wang and Lijuan Liu
increasing on (0, ∞), we obtain
Z
Z
ρL (u)
ρL (u)
1
e
log
log
dVi,K =
ρn−i
K (u) dS(u)
f
ρ
(u)
ρ
(u)
n−1
n−1
K
K
nWi (K) S
S
!
Z
ρL (u) n−i
1
≤ log
ρ (u) dS(u)
fi (K) S n−1 ρK (u) K
nW
!
fi (K, L)
W
= log
(3.6)
fi (K)
W
n−i−1
1 !
fi (L) n−i
fi (K) n−i W
W
≤ log
fi (K)
W
! 1
fi (L) n−i
W
= log
.
fi (K)
W
This gives the desired inequality. Since log(·) is strictly increasing, from the equality
conditions of the dual Minkowski inequality (2.8), we have that equality in (3.6) holds if
and only if K and L are dilates.
Remark 3.8. The case i = 0 of Theorem 3.6 and Theorem 3.7 are Theorem 1.5 and
Theorem 1.6, respectively.
Lemma 3.9. Let K, L ∈ S n . Then
lim
ρ(1−λ)·K +
e 0 λ·L (u) − ρK (u)
λ
λ→0+
= ρK (u) log
ρL (u)
ρK (u)
,
uniformly for all u ∈ S n−1 .
Proof. For u ∈ S n−1 , we have
lim
λ→0+
ρ(1−λ)·K +
e 0 λ·L (u) − ρK (u)
λ
ρL (u)
ρK (u)
λ
−1
= ρK (u) lim
λ
λ→0+
ρL (u)
= ρK (u) log
.
ρK (u)
Then the pointwise limit has been proved. Moreover, the convergence is uniform for any
u ∈ S n−1 .
Lemma 3.10. Let K, L ∈ S n . For i = 0, 1, . . . , n − 1, then
Z
fi (K)
fi ((1 − λ) · K +
e 0 λ · L) − W
W
n−i
ρL (u)
lim
=
log
ρn−i
K (u) dS(u).
λ
n
ρK (u)
λ→0+
S n−1
The Dual Log-Brunn-Minkowski Inequalities
917
Proof. By (2.6) and Lemma 3.9, it follows that
lim
λ→0+
= lim
λ→0+
fi ((1 − λ) · K +
fi (K)
e 0 λ · L) − W
W
λ
n−i
n−i
Z
ρ
e λ·L (u) − ρK (u)
1
(1−λ)·K +
0
n
S n−1
λ
n−i
(u) − ρK
(u)
λ·L
dS(u))
Z
ρn−i
e0
1
(1−λ)·K +
lim
dS(u)
=
+
n S n−1 λ→0
λ
Z
ρK +
n−i
e 0 λ·L (u) − ρK (u)
ρn−i−1
=
(u) lim
dS(u)
K
+
n
λ
λ→0
S n−1
Z
ρL (u)
n−i
n−i
log
ρK
(u) dS(u).
=
n
ρ
(u)
n−1
K
S
Theorem 3.11. Let K and L be two star bodies in Rn . Then the dual log-BrunnMinkowski inequality and the dual log-Minkowski inequality are equivalent.
e 0 λ · L, it is obviously that Q0 = K, Q1 = L. We will first
Proof. Let Qλ = (1 − λ) · K +
suppose that we have the dual log-Minkowski inequality (3.5). For 0 < λ < 1, by (3.5),
we have
Z
ρK (u)1−λ ρL (u)λ
dVei,Qλ
0=
log
ρQλ (u)
S n−1
Z
Z
ρK (u)
ρL (u)
e
= (1 − λ)
log
dVi,Qλ + λ
log
dVei,Qλ
ρQλ (u)
ρQλ (u)
S n−1
S n−1
fi (K)
fi (L)
W
λ
W
1−λ
log
+
log
≤
fi (Qλ ) n − i
fi (Qλ )
n−i
W
W
fi (K)1−λ W
fi (L)λ
1
W
=
.
log
fi (Qλ )
n−i
W
This gives the dual log-Brunn-Minkowski inequality (3.2). From the equality conditions
of the dual log-Minkowski inequality (3.5), we have that equality in (3.2) holds if and only
if K and L are dilates.
Suppose now that the dual log-Brunn-Minkowski inequality (3.2) holds. We define the
fi (Qλ ).
function f : [0, 1] → (0, ∞) by f (λ) = W
For given σ, τ ∈ [0, 1], if α ∈ [0, 1] and α = (1 − λ)σ + λτ , we have
1−λ
ρ(1−λ)·Kσ +
ρKτ (u)λ
e 0 λ·Kτ = ρKσ (u)
1−λ
λ
= ρK (u)1−σ ρL (u)σ
ρK (u)1−τ ρL (u)τ
= ρK (u)1−[(1−λ)σ+λτ ] ρL (u)(1−λ)σ+λτ
= ρK (u)1−α ρL (u)α
= ρQα .
918
Wei Wang and Lijuan Liu
Thus,
e 0 λ · Kτ = (1 − α) · K +
e 0 α · L.
(1 − λ) · Kσ +
(3.7)
From (3.7) and (3.2), we have
fi ((1 − α) · K +
e 0 α · L)
f (α) = f ((1 − λ)σ + λτ ) = W
fi ((1 − λ) · Kσ +
e 0 λ · Kτ )
=W
fi (Kσ )1−λ W
fi (Kτ )λ
≤W
= f (σ)1−λ f (τ )λ ,
fi (Qλ ) is a
which is the desired log convexity of f . Equivalently, the function λ 7→ log W
convex function, and thus
(3.8)
fi (Qλ ) dW
fi (Q1 ) − log W
fi (Q0 ) = log W
fi (L) − log W
fi (K).
≤ log W
fi (Q0 )
dλ λ=0
W
1
Combining (3.8) and Lemma 3.10, we obtain the dual log-Minkowski inequality. From
the equality conditions of the dual log-Brunn-Minkowski inequality (3.2), we have that
equality in (3.5) holds if and only if K and L are dilates.
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The Dual Log-Brunn-Minkowski Inequalities
919
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http://dx.doi.org/10.1017/cbo9780511526282
Wei Wang
Department of Mathematics, Tongji University, Siping Road, Shanghai, 200092,
P. R. China
and
School of Mathematics and Computational Science, Hunan University of Science and
Technology, Xiangtan, 411201, P. R. China
E-mail address: [email protected]
Lijuan Liu
School of Mathematics and Computational Science, Hunan University of Science and
Technology, Xiangtan, 411201, P. R. China
E-mail address: [email protected]