Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
DOI 10.1186/s13660-015-0572-0
RESEARCH
Open Access
Generalized Finsler-Hadwiger type
inequalities for simplices and applications
Shiguo Yang1,2 and Wen Wang1*
*
Correspondence:
[email protected]
1
School of Mathematics and
Statistics, Hefei Normal University,
Hefei, Anhui 230601, P.R. China
Full list of author information is
available at the end of the article
Abstract
For an n-dimensional simplex in the Euclidean space E n , we establish some
generalized Finsler-Hadwiger inequalities. By using these inequalities, generalizations
of some well-known important inequalities for simplices are obtained.
MSC: 51M20; 52A40
Keywords: simplex; volume; circumradius; inradius; inequality
1 Introduction
Let ABC be a triangle (in the Euclidean plane) of area S and sides a, b, c. The following
inequality [, ] is well known:
√
a + b + c ≥  S + (a – b) + (b – c) + (c – a) .
(.)
Equality holds iff this triangle is regular.
Let n be an n-dimensional simplex (in the n-dimensional Euclidean space En ) of volume V and edge lengths ai (i = , , . . . ,  n(n + )). Let Ai (i = , , . . . , n) be the vertices of
n , Fi be the (n – )-dimensional volume of the ith face fi = A · · · Ai– Ai+ · · · An+ ((n – )dimensional simplex) for i = , , . . . , n. In , Chen and Ma extended the inequality (.)
to an n-simplex and established Finsler-Hadwiger type inequality for the edge length and
the volume of an n-simplex as follows [, ]:

 n(n+)
ai ≥ n(n + )
i=
(n!)
n+
n

Vn +

n(n – )
(ai – aj ) .
(.)
≤i<j≤  n(n+)
Equality holds iff n is regular.
In , Leng and Tang established another kind of Finsler-Hadwiger type inequality for
the face areas and the volume of an n-simplex as follows [, ]:
n
i=

Fi ≥
n (n + ) n
(n!)

n
V
(n–)
n
+

(Fi – Fj ) .
n –  ≤i<j≤n
(.)
Equality holds iff n is regular.
In this paper, we will discuss problem for generalized Finsler-Hadwiger type inequalities for k-dimensional face areas and the volume of an n-simplex, and establish generalized
© 2015 Yang and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 2 of 14
Finsler-Hadwiger type inequalities for an n-simplex. From these inequalities, generalizations of some inequalities for an n-simplex are obtained.
2 Main results
Let R and r be the circumradius and the inradius of the simplex n , respectively. Let
V denote the volume of the n-dimensional regular simplex with circumradius R. For
(n+)!
k = , , . . . , n – , we put μn,k = n+
= (k+)!(n–k)!
. Let Vi (k) (i = , , . . . , μn,k ) denote the
k+
k-dimensional volumes of the k-dimensional subsimplices of n . Our main results are the
following theorems.
Theorem  Let n be an n-simplex, nature number k ∈ [, n – ], real numbers α ∈ (, ]
and λ ∈ (, n +  – k]. Then we have
μn,k
Viα (k) ≥
i=
+
R
nr
kα
n(n+)(n–)

k k +  (n!) n α kα
· μn,k ·
V n
(k!) n + 
μn,k – λ ≤i<j≤μ
α

Vi (k) – Vjα (k) .
(.)
n,k
Equality holds iff n is regular.
Theorem  Let n be an n-simplex, nature number k ∈ [, n – ], real numbers α ∈ (, ]
and λ ∈ (, n +  – k]. Then we have
μn,k
Viα (k) ≥
i=
+
V
V
kα
n(n+)(n–)

k k +  (n!) n α kα
· μn,k ·
V n
(k!) n + 
μn,k – λ ≤i<j≤μ
α

Vi (k) – Vjα (k) .
(.)
n,k
Equality holds iff n is regular.
R
≥  and VV ≥ , respecNotice From the inequality R ≥ nr and Lemma , we have nr
tively. Further, according to the conditions of equalities hold, we now consider a class of
n-simplices ϕ inscribed in an (n – )-dimensional sphere of radius R (where R is constant
R
number). If the radius r of an n-simplex in ϕ is sufficiently small, then nr
is large enough.
Similarly, if the volume V of an n-simplex in ϕ is sufficiently small, then VV is sufficiently
large.
Take λ = n +  – k, from Theorem  or Theorem  we derive an inequality for a simplex
as follows.
Corollary  Let n be an n-simplex and real number α ∈ (, ]. Then
μn,k
k k +  (n!) n α kα
·
V n
(k!) n + 
Viα (k) ≥ μn,k
i=
+


Viα (k) – Vjα (k) .
μn,k – (n +  – k) ≤i<j≤μ
n,k
Equality holds iff n is regular.
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
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By taking k =  and α =  in the inequality (.) we can obtain the inequality (.). Take
k = n –  and α =  in the inequality (.) we can obtain the inequality (.).
Take k = , α =  and λ = n in Theorem  and Theorem , we get generalizations of the
inequality (.) as follows.
Corollary  For an n-simplex n , we have

 n(n+)
ai
≥
i=
ai ≥
i=
V
V
+

n(n+)(n–)

n(n – )
+

 n(n+)
R
nr
(n!)
· n(n + ) ·
n+

Vn
(ai – aj ) ,
(.)
≤i<j≤  n(n+)

n(n+)(n–)

n(n – )
n
· n(n + ) ·
(n!)
n+
n

Vn
(ai – aj ) .
(.)
≤i<j≤  n(n+)
Equality holds iff n is regular.
Take k = n – , α = , and λ =  in Theorem  and Theorem ; we get generalizations of
the inequality (.) as follows.
Corollary  Let n be an n-simplex. Then we have
n
Fi
≥
i=
n
Fi
i=
≥
R
nr
V
V

n(n –)

n(n –)
V
(n–)
n
+

(Fi – Fj ) ,
n –  ≤i<j≤n
(.)
V
(n–)
n
+

(Fi – Fj ) .
n –  ≤i<j≤n
(.)

·
n (n + ) n
(n!)

n

·
n (n + ) n
(n!)

n
Equality holds iff n is regular.
To prove the above theorems, we need some lemmas as follows.
Lemma  [, ] For an n-simplex n , we have
n
i=
n–
Fi
n(n–)
≥
(n + )n– (n!)
n+

V
n –n–
 n(n+) n

ai ,
(.)
i=
with equality iff n is regular.
Lemma  [, ] Let n be an n-simplex with edge lengths aij = |Ai Aj | ( ≤ i < j ≤ n), then
we have
n+ n

n
n
aij ≥
(n!)  (VR)  ,
(.)
n
≤i<j≤n
with equality iff n is regular.
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
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Lemma  [, ] For an n-simplex n , we have
V≤
(n + )n+
(n!) nn

Rn ,
(.)
with equality iff n is regular.
Lemma  [, ] For an n-simplex n , we have
n
n–
Fi



≥ nn – (n + )–(n+)(n–)  (n!)–n V n –n– R.
(.)
i=
Equality holds iff n is regular.
Lemma  [, ] Let n be an n-simplex and natural number k ∈ [, n – ). Then
√
μn,k
k!
k+
μ
n,k
Vi (k)
i=
 k
n n+
n–
(n – )! ≥
Fi
,
√
n
i=
(.)
with equality iff n is regular.
Lemma  For an n-simplex n , we have
n+
(n + )  n
V≥
n!
n –
n
r
n –
n

Rn ,
(.)
with equality iff n is regular.
Proof Let P be an interior point of n and di the distance from the point P to the ith face
fi of n for i = , , . . . , n. Obviously, we have ni= di Fi = nV . Then
n
di Fi
i=
nV
= .
Using the arithmetic-geometric means inequality we get
n
di Fi
i=
nV
≤
 di Fi
n +  i= nV
n
n+
=

,
(n + )n+
namely
n
n–

≤
di
i=
(nV )n –
.
(n + )n – ( ni= Fi )n–
(.)
Substituting (.) into (.) we get
n
i=
n–
di
n
 –  V
.
≤ (n!)n (n + )n(n+) nn – 
R
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
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We now take the point P is the incenter of n , then di = r for i = , , . . . , n. Thus from
(.) we can obtain the inequality (.). It is easy to see that equality holds in (.) iff
n is regular.
Lemma  For an n-simplex n and k = , , . . . , n – , we have
μn,k
μ
n,k
Vi (k)
≥
i=
μn,k
μ
n,k
Vi (k)
≥
i=
R
nr
V
V
√
k
n(n+)(n–)
k
n(n+)(n–)
nk
k+
n!
V ,
√
k!
n+
√
nk
n!
k+
V .
√
k!
n+
(.)
(.)
Equality holds iff n is regular.
Proof We have the well-known inequality [, , ]
V≤
(n + )
n
n–

n+

Rn– r.
(.)
n!
Equality holds iff n is regular.
Using (.) and (.) we get
n
n–
Fi
i=

nn –
≥
(n + )n –n–

V
(n –)(n–)
n
(n!n )V
R

n
.
(.)
From (.) and (.) we have
n

n+
≥
Fi
i=
R
nr

n(n –)

n

n
(n!) (n + )
n–
n
V
n–
n
.
(.)
Thus, inequality (.) is valid for k = n – . We now prove that inequality (.) is valid
for  ≤ k < n – . Substituting (.) into (.) we can obtain inequality (.). So the
inequality (.) is valid for  ≤ k < n – . It is easy to see that equality holds in (.) iff n
is regular.
By Lemma  we have
V =
(n + )n+
(n!) nn

Rn .
(.)
Substituting (.) into the right of (.) we get

 n(n+)
ai ≥
i=
V
V
 n
n+
n+

(n!V )
n+

.
(.)
From (.) and (.) we have
n
i=

n+
Fi
≥
V
V

n(n –)

n

n
(n!) (n + )
n–
n
V
n–
n
.
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
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Thus the equality (.) is valid for k = n – . Now we prove that inequality (.) is valid
for  ≤ k < n – . Substituting (.) into the right of (.) we can obtain inequality (.).
So the inequality (.) is valid for  ≤ k < n – . It is easy to see that equality holds in (.)
iff n is regular.
Lemma  [, ] Let n be an n-simplex and real number α ∈ (, ]. Then

Fiα Fjα
–
≤i<j≤n
n
Fiα
i=
  n n +  n α (n–)α
≥ n –
V n ,
n +  (n!)
(.)
with equality iff n is regular.
Proof of Theorem  and Theorem  The inequality (.) can be written as
μn,k
Viα (k) + 
Viα (k)Vjα (k) – λ
≤i<j≤μn,k
i=
≥
R
nr
μn,k
i=
√
kα
n(n+)(n–)
Viα (k)
· μn,k (μn,k – λ) ·
nk α
n!
k+
kα
V n .
√
k!
n+
(.)
Now we prove that the inequality (.) is valid.
Let p(k, α, δ) denote the left of the inequality (.). Then
p(k, α, δ) = 
Viα (k)Vjα (k) – (n – k)
≤i<j≤μn,k

μn,k
Viα (k)
i=
Viαp (k)Viαq (k) –
k+
Viα
(k)
p
p=
≤p<q≤k+
(i ,i ,...,ik+ )∈σ
+
Viα (k) + (n +  – k – λ)
i=
=
μn,k
Viαp (k)Viαq (k) + (n +  – k – λ)
ip ,iq ∈τ
μn,k
Viα (k),
(.)
i=
where σ = {(i , i , . . . , ik+ )| there exists a (k + )-subsimplex Ai Ai · · · Aik+ of n , such that
its k + side areas are Vi (k), Vi (k), . . . , Vik+ (k) ( ≤ i < i < · · · < ik+ μn,k )}, and τ = {(ip , iq )|
there is not a (k + )-subsimplex of n , such that its two side areas are Vip (k), Viq (k)}.
On the other hand, we easily get
n+
μn,k
n+

|σ | =
,
|τ | =
–
= μn,k – (n – k)(k + ) –  .
k+

k+

n+ n–k
We put m = |σ |, then m = k+
= k+ · μn,k . If (i , i , . . . , ik+ ) ∈ σ , we use Vi (k + ) to denote
the volume of k + -subsimplex with side areas Vi (k), Vi (k), . . . , Vik+ (k). By Lemma  we
have

Viαp (k)Viαq (k) –
k+
Viα
(k)
p
p=
≤p<q≤k+
(k + )
≥ k(k + )
k+

k+
(k + )!

k+
α
Vi (k + )
kα
Equality holds iff the simplex Ai Ai · · · Aik+ is regular.
k+
.
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
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Using the arithmetic-geometric means inequality, (.), and (.) we get
I =

Viαp (k)Viαq (k) –
k(k + )
(i , ,...,ik+ )∈σ
k+
Viα
(k)
p
p=
≤p<q≤k+
(i ,i ,...,ik+ )∈σ
≥
 α
k+
kα
k+
(k + )
Vi (k + ) k+

k +  (k + )!
kα
 α μ
n,k+
(k+)μn,k+
k+
k+
(k + )
≥ μn,k+ k(k + )
V
(k
+
)
i
k +  (k + )!
i=
 α
k+
(k + )
k+
≥
k(k + )
k +  (k + )!
√
k+
kα
n
k+
n!
k+
kα
· μn,k+
V n
√
(k + )!
n+
kα 
√
nk α
R n(n+)(n–)
n!
k+
kα
=
μn,k+ k(n – k)
V n .
√
nr
(k)!
n+
R
nr
kα
n(n+)(n–)
(.)
By the arithmetic-geometric means inequality, and by (.), we get
I =
Viαp (k)Viαq (k) + (n +  – k
– λ)
μn,k
Viα (k)
i=
(ip ,iq )∈τ
≥ μn,k μn,k – (n – k)(k + ) – 
μn,k
μα
n,k
Vi (k)
+ (n +  – k – λ)μn,k
μn,k
i=
≥
R
nr
kα
n(n+)(n–)
μα
n,k
Vi (k)
i=
μn,k μn,k – (n – k)k – λ
√
k+
n!
√
(k)!
n+
nk α
V
kα
n
.
(.)
From (.), (.), and (.) we get
p(k, α, λ) = I + I
≥
R
nr
kα
n(n+)(n–)
√
μn,k [μn,k – λ]
nk α
k+
n!
kα
V n .
√
(k)!
n+
(.)
Thus the equality (.) is valid. It is easy to prove that the equality holds iff n is regular.
Similarly, we can prove that (.) is true by the same method.
3 Some applications
Geometric inequalities for simplices which are the simplest and useful polytopes have
been a very attractive subject for a long time. Mitrinović, Pečarić, Volenec, Zhang and
Yang and other authors have obtained a great number of elegant results (see []). Specially,
we mention the well-known Euler inequality for a simplex, inequalities for the width of a
simplex, and inequalities related the interior point of a simplex. In this section we shall
improve these inequalities by using the results in the above section.
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 8 of 14
LF Tóth extended the Euler inequality R ≥ r for a triangle to an n-dimensional simplex
and obtained the Euler inequality for an n-simplex as follows [, ]:
R ≥ nr.
(.)
Equality holds iff n is regular.
Let O and I denote the circumcenter and the incenter of an n-simplex n , respectively.
Let G be the barycenter of n . Klamkin [, ] and Yang and Wang obtained the following
generalizations of (.):
R – |OI| ≥ (nr) ,
(.)
R – |OG| ≥ (nr) .
(.)
Equality holds iff n is regular.
We shall use these results in the above section to study related inequalities, and we obtain
a strengthened Euler inequality as follows.
Theorem  For an n-simplex n and k = , , . . . , n – , we have
R – |OG|
k
≥
R
nr
k
n(n+)(n–)
(nr)k +


Vi (k) – Vj (k) ,
μn,k – (n +  – k) ≤i<j≤μ
n,k
(.)
R – |OG|
k
≥
V
V
k
n(n+)(n–)
(nr)k +


Vi (k) – Vj (k) .
μn,k – (n +  – k) ≤i<j≤μ
n,k
(.)
Equality holds iff n is regular.
If take k =  and k = n –  in (.) and (.), we get two corollaries as follows.
Corollary  For an n-simplex n and k = , , . . . , n – , we have
R – |OG| ≥


R – |OG| ≥
R
nr
V
V

n(n+)(n–)

n(n+)(n–)
(nr) +
(nr) +

n(n – )

n(n – )
(ai – aj ) ,
(.)
(ai – aj ) ,
(.)
≤i<j≤  n(n+)
≤i<j≤  n(n+)
with equality iff n is regular.
Corollary  For an n-simplex n and k = , , . . . , n – , we have
 n–
R – |OG|

≥
R
nr

n(n –)
(nr)(n–) +

(Fi – Fj ) ,
n –  ≤i<j≤n
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
 n–
R – |OG|

≥
V
V

n(n –)
Page 9 of 14
(nr)(n–) +

(Fi – Fj ) .
n –  ≤i<j≤n
(.)
Equality holds iff n is regular.
Proof of Theorem  If take α =  and λ = n +  – k in (.) and (.), we obtain
μn,k
R
nr
Vi (k) ≥
i=
+

n(n+)(n–)
k
k +  (n!) n k
μn,k ·
Vn
(k!) n + 


Vi (k) – Vj (k) ,
μn,k – (n +  – k) ≤i<j≤μ
(.)
n,k
μn,k
V
V
Vi (k) ≥
i=
+

n(n+)(n–)
k
k +  (n!) n k
μn,k ·
Vn
(k!) n + 


Vi (k) – Vj (k) .
μn,k – (n +  – k) ≤i<j≤μ
(.)
n,k
We use the well-known equality [, ]

 n(n+)
ai = (n + ) R
(.)
i=
and the well-known inequality [, ]
μn,k
Vi (k) ≤
i=
n!
k

n (k!) (n – k)!(n + )k–
 n(n+)

k
ai
.
(.)
i=
Equality holds iff n is regular. When k = , then (.) is the identity.
By (.) and (.) we get
μn,k
Vi (k) ≤
i=
(n + )k+ n! k
R .
(k!) nk (n – k)!
(.)
We use the well-known inequality in []
n
V≥
n  (n + )
n!
n+

rn ,
(.)
with equality iff n is regular.
From (.), (.), and (.) we can obtain (.). By (.), (.), and (.) we can
obtain (.). It is easy to see that equality holds in (.) or (.) iff n is regular.
Let ω be the width of n . We put
βn =
n[ n+
](n +  – [ n+
])


.
(n + )
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 10 of 14
In , Alexander [, ] proved Sallee’s conjecture and obtained the inequality
ω– ≥ βn R– .
(.)
Equality holds iff n is regular.
In , Yang and Zhang [, ] improved (.) and established an inequality as follows:
ω– ≥ βn
(n + )
n+
n
n · (n!)

n

V–n ,
(.)
with equality iff n is regular.
In this section, we shall give generalizations of the equalities (.) and (.) by using
(.) and (.) in the above section.
Theorem  For an n-simplex n and k = , , . . . , n – , we have
ω– ≥
ω
–
≥
R
nr
V
V

n(n –)

n(n –)
βn
βn
(n + )
n+
n
n · (n!)
(n + )

n
n+
n

n · (n!) n

V – n + βn

V – n + βn
(n + )V – (Fi – Fj ) ,
n (n – ) ≤i<j≤n
(.)
(n + )V – (Fi – Fj ) .
n (n – ) ≤i<j≤n
(.)
Equality holds iff n is regular.
Proof We use the well-known inequality in [] as follows:
n (n + )
ω ≤ n+
[  ](n +  – [ n+
])


(
n
V
 
i= Fi )
.
Equality holds iff n is regular.
The above inequality can be written as
ω
–
n+
≥ βn · 
n
n

i= Fi
.
V
(.)
By (.) and (.) we can get (.), and from (.) and (.) we can get (.).
Substituting (.) into (.) and (.) we obtain generalizations of the inequality (.)
as follows.
Corollary  For an n-simplex n and k = , , . . . , n – , we have
ω– ≥
ω
–
≥
R
nr
V
V

n(n –)

n(n –)
βn R– + βn
(n + )V – (Fi – Fj ) ,
n (n – ) ≤i<j≤n
(.)
βn R– + βn
(n + )V – (Fi – Fj ) .
n (n – ) ≤i<j≤n
(.)
Equality holds iff n is regular.
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 11 of 14
For i = , , . . . , n, let hi be the altitude of the ith face of the simplex n . The following
inequality is well known (see []):
n
n n+
n
n
V ≥
hi
.
(n!) (n + )n– i=
(.)
Equality holds iff n is regular.
By using the results in Section  we can obtain generalizations of the inequality (.)
as follows.
Theorem  For an n-simplex n and k = , , . . . , n – , we have
V
(n –)k
R
nr
≥
n (n –)k
n n+
nn
·
hi
(n!) (n + )n– i=
kα
n(n+)(n–)



(nr)(n –n–)k
(n + )n+ (n –)k Vi (k) – Vj (k) ,
+
n

μn,k – (n +  – k) n · (n!)
≤i<j≤μ
(.)
n,k
V
(n –)k
V
V
≥
+
n (n –)k
n n+
nn
·
hi
(n!) (n + )n– i=
kα
n(n+)(n–)



(n + )n+ (n –)k (nr)(n –n–)k
Vi (k) – Vj (k) .
n

μn,k – (n +  – k) n · (n!)
≤i<j≤μ
(.)
n,k
Equality holds iff n is regular.
Proof We take k = n –  in (.) and (.), and get
n
Fi ≥
i=
n
Fi ≥
i=
R
nr
V
V

n(n–)
(n!)
V
n+
n

n(n–)
Using the formula Fi =
(n –)k
n
(n+)

(n + )
n
(n!)
nV
hi
n+
n
n –
n
(n+)

(n + )
n –
n
V
n –
n
,
(.)
V
n –
n
.
(.)
(see []) and (.) we get
n (n –)k
n n+
nn
Rk
≥
hi
.
(nr)k (n!) (n + )n– i=
(.)
By Theorem  we get
Rk ≥
R
nr
k
n(n+)(n–)
(nr)k +


Vi (k) – Vj (k) .
μn,k – (n +  – k) ≤i<j≤μ
n,k
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 12 of 14
Substituting (.) into (.) we get
V
(n –)k
≥
R
nr
kα
n(n+)(n–)
n (n –)k
n n+
nn
·
hi
(n!) (n + )n– i=
n (n –)k
n n+
nn

·
+
hi
(nr)k [μn,k – (n +  – k)]
(n!) (n + )n– i=

Vi (k) – Vj (k) .
·
(.)
≤i<j≤μn,k
We use the well-known inequality in [, ] as follows:
n
hi ≥ (n + )n+ rn+ ,
(.)
i=
with equality holding iff n is regular.
From (.) and (.) we can obtain (.). It is easy to see that equality holds iff n is
regular.
By a similar method we can prove that inequality (.) is also valid.
Let P be an interior point of n and di the distance from the point P to the ith face fi of
n for i = , , . . . , n. In [], Gerber established the following inequality:
n –  
nn(n+) (n + )(n+) n – (n+)
di
≥
V n .
(.)
(n!)(n+)
i=
For any natural number m > , Leng and Ma [] obtained an inequality as follows:
(di di )–m – (μn, )–m ·  · r–m
mn
n
≥ (μn, )–m (μn, )m –  nm R–m m .
(.)
≤i<j≤n
Equalities hold in (.) and (.) iff n is regular and P is the center of n .
By using the results in the Section  we can obtain generalizations of (.) and (.)
as follows.
Theorem  For an n-simplex n , we have
n – 
n+
 
R n(n –)
n (n + )(n+) n – (n+)
di
≥
·
V n
nr
(n!)(n+)
i=
+
n
–
di
≥
i=
+
nn– (n + )n+ V – (Fi – Fj ) ,
(n!) (n – )
≤i<j≤n
V
V

n(n –)
nn+ (n + )(n+)
·
(n!)(n+)

n
V–
(.)
(n+)
n
nn– (n + )n+ V – (Fi – Fj ) .
(n!) (n – )
≤i<j≤n
Equality holds in (.) or (.) iff n is regular and P is the center of n .
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 13 of 14
Proof By (.), we have
n
–
(n + )(n+) –(n+) 
V
Fi .
n(n+)
i=
n
≥
di
i=
(.)
We use the well-known inequality in [] as follows:
n
nn
(n–)
≥
V
Fi .
(n!) (n + )n
i=
n
Fi
i=
(.)
Equality holds iff n is regular.
Using (.) and (.) we get
n
Fi
≥
i=
+
R
nr

n(n –)
n(n+)
(n!)
(n+)
n
(n + )
n –
n
nn V (n–)
(n!) (n + )n (n – )
V
(n –)
n
(Fi – Fj ) .
(.)
≤i<j≤n
From (.) and (.) we can obtain (.). It is easy to see that equality holds in (.)
iff n is regular and P is the center of n .
By a similar method we can prove that (.) is valid.
Theorem  Let n be an n-simplex and m >  a natural number, then we have
(di dj )–m – (μn, )–m r–m
mn
≤i<j≤n
≥
R
nr

n(n+)(n–)
n
(μn, )–m (μn, )m –  nm R–m m
· γ (n, m)(Rr)–
(ai – aj ) ,
(.)
≤i<j≤μn,
(di dj )–m – (μn, )–m r–m
mn
≤i<j≤n
≥
V
V

n(n+)(n–)
n
(μn, )–m (μn, )m –  nm R–m m
· γ (n, m)(Rr)–
(ai – aj ) .
≤i<j≤μn,
Equality holds iff n is regular and P is the center of n , where
n
γn,m = (μn, )–m (μn, )m –  m ·

.
nn+ (n – )
Proof We use the following well-known inequality (.) in []:
≤i<j≤n
(di dj )
–m
≥ (μn, )
–m
mn
(μn, )m – 
–m
m m n+
+ (n + ) n
·
r
,
m
n!
V n
(.)
Yang and Wang Journal of Inequalities and Applications (2015) 2015:56
Page 14 of 14
i.e.
(di dj )–m – (μn, )–m r–m
≤i<j≤n
≥ (μn, )
–m
m m
(n + ) n
m 
n+ · (μn, )m –  n  .

n!
V
(.)
Equality holds iff n is regular and P is the center of n .
From (.) we have
n! · nn

≥
R .

V
(n + )n+ Rn (nr)
(.)
Substituting (.) into (.) and using (.) we can obtain (.). It is easy to see that
equality holds in (.) iff n is regular and P is the center of n .
The proof of (.) is similar.
In fact, the strengthening of some well-known inequalities for simplices can be derived
from Theorem  and Theorem . In this paper, we omit the details.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors co-authored this paper together. All authors read and approved the final manuscript.
Author details
1
School of Mathematics and Statistics, Hefei Normal University, Hefei, Anhui 230601, P.R. China. 2 Department of
Mathematics and Physics, Anhui Xinhua University, Hefei, 230088, P.R. China.
Acknowledgements
This work is supported by the Doctoral Programs Foundation of Education Ministry of China (20113401110009) and
Foundation of Anhui higher school (KJ2013A220); Natural Science Research Project of Hefei Normal University (2012kj11).
The authors are grateful for the help.
Received: 6 July 2014 Accepted: 22 January 2015
References
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