Advances in Pure Mathematics, 2011, 1, 264-266
doi:10.4236/apm.2011.15046 Published Online September 2011 (http://www.SciRP.org/journal/apm)
A Volume Product Representation and Its
Ramifications in ,1
n
p
lp


Dimitris Karayannakis
Department of Sci ence , Division of Mathematics, TEI of Crete, Heraklion, Greece
Received April 19, 2011; revised June 9, 2011; accepted June 20, 2011
Abstract
Let n
p
B
, , be the volume of the unit 1p
p
-ball in n
R
and q the Hölder conjugate exponent of
p
. We represent the volume product n
p
B
n
q
B
as a function free of its gamma symbolism. This representa-
tion will allows us in this particular case to confirm, using basic classical analysis tools, two conjectured and
partially proved lower and upper bounds for the volume product of centrally symmetric convex bodies of the
Euclidean n
R
. These bounds in the general case play a central role in convex geometric analysis.
Keywords: P-Ball, Volume, Gamma Function, Infinite Product
1. Introduction
One of the key notions in convex analysis is the volume
product

o
M
KKK where
K
is a (centrally sym-
metric) convex compact set of the Euclidean


,.,.
n
R
with non-empty interior (or simply “body”) and

=:,1,
on
K
yR xyxK is the so called polar
set of
K
For this product, one of the long standing conjectures
stated by Mahler (who proved it for n = 2) claims that

4
!
n
M
K
nfor origin symmetric bodies . This con-
K
jecture has been confirmed in many special cases of
K
and in particular for bodies symmetric with respect to the
coordinate planes, which naturally include the -balls,
using a rather advanced Banach space theory. On the
other hand, for an upper bound, we have the inequality
p


2
n
M
KMB
=2n proved in 1948 by Santaló (and much earlier for by Blashke). A survey of the above
facts and other related results can be found e.g. in [4].
We set forward in this short work to establish
Mahler’s conjecture and the Blashke-Santaló inequality
for the case using exclusively basic
special functions and classical analysis theory.
=,1
n
p
KB p,
n
It is clear that

=,
o
n
p
q
BB where 11
=1
pq
, and
also directly verifiable through multiple integration (see
e.g. [1]) that
1
21
1
n
n
n
p
p
Bn
p








for and
1p
1=2
nn
BB

n .
So we now have to manipulate the expression

11
11
=4 .
11
n
nn
p
pq
MB nn
pq



 

 




 


(1)
Thus, in Sec.2 we start with a suitable for our goals
gamma functions ratio result (Lemma) that will allow us to
represent


n
p
M
B as a suitable function


,,
M
np for
1p

, free of its gamma symbolism (Proposition);
finally, in Sec.3, by establishing

d, 0
d
Mnp
p for
1< 2p

(Lemma) and then using the evident facts that




,,=
M
npM nq and 1 iff , and
also by examining separately the case , we will
2q 2<p
1p
obtain the announced results (Proposition). We also ob-
tain, as byproducts of independent interest, three seem-
ingly new closed formulae concerning infinite products
(Corol. 1, Corol. 2 and Rem. 2).

D. KARAYANNAKIS 265
2. A Lemma and a Proposition
Lemma
For and
>0x0< 1

 we have that




 
=1
1
1= 1
k
xkkx
xkkx



 
 
.


(2)
Proof:
Let

,Px

be the infinite product in lemma’s
statement .One of the classical definitions for the gamma
function for any complex , is 0, 1,2,z
 
1!
=,
lim
z
kk
kk
zz



where is the shifted factorial
.Substituting in the above limit,
respectively,

k
z

1

zzz k
=1

1
, =,zzx

 and =
z
x, and after
simplifying we arrive at




 
!
1=
lim 1
k
kkk
kx
x
xx



 

 
which is iden-
tical to

,Px

.
Remark
1) Independently of the above argumentation we can
easily check that

,Px


exists as a two-variable func-
ction over since evidently
  
0,10,


0,Px


0 and, by use of the inequality , log1, >tt t


 
=1
log ,
1
1
1
k
Px
xkkx

 

 

<
.
2) This lemma was proved formally (and from “scratch”)
in [3] where the scheme of the proof served different
purposes concerning the numerical evaluation of the
gamma function.
Proposition
For ,
1< <p


2
2
22
22
22
=1
,=
1
4(1)
n
n
nn
k
Mnp
n
knk kk
pq pq
hp k






 
 




(3)
where

π
=π
sin
hp
pq p



.
Proof:
Evidently and so (2) is trivially true for
. Thus we can consider
(1)= 1h
=1n2.n
At first let us rewrite (1) as
 
21
11
41
,
n
n
n
pq
Mnp nnn
pq pq




 






Based on Lemma II.1 for 1 =p  and
=,=1,, 1
m
x
mn
p

 we see that
1
1
=1
1
1
=,
1
n
nm
p nn
P
pp
q




 


 
 



m
p
(4)
where


,Px

was defined above in Lemma
Working in a similar way we obtain the “conjugate
expression”


.nq
We observe now that

  
2
2
=1
11
,,
1,
1
=11
k
kkk
nm nm
PP
pp qq
krnmp
nm
pqn mgprn mp

 






 

,
(5)
where we have set

2
2
,=
kj
rjp kjkpq  and




=1,
kk
g
pr p.
Combining (2), (4), and (5), and by “telescoping”, we
obtain (3).
3. One More Lemma and a Conclusive
Proposition
Lemma
Let 1p2

. Then

d, 0
d
Mnp
p for all n with
equality iff for =2p1n

.
Proof:


1,= 4n Mp
p
and once more we can consider
By straightforward differentiation with respect to
and in the case of the infinite product of (2) by logari-
thmic differentiation (noticing that this product by con-
struction is a real analytic function of having as
logarithm a uniformly converging series of differentiable
functions of ) we obtain
2.np
p

2
=1
d(,)
d
12
=22
kkk
Mnp
sign p
nn
sign nhh
pqg r




.



 



 




 



(6)

D. KARAYANNAKIS 266
In (6) was defined in (3) and
k

=hhp

=,=
k
 
,
kk
g
gpr r
p
np
n
as defined in (5); still for sim-
plicity from now on we suppress any possible depen-
dence from and .
Since evidently and the infinite sum in (6) are h
positive and
 

2
2
=
1
pp
pq p


0
(with equality iff
=2pp) it will suffice to show that (with equality
iff ).
0h
=2
Simple calculations show that

π
sinsignhsign pqsign
p




 



 , where
 
π
==2sin π1cos .pp pp
pp

 

 
 
π
(7)
Now
 

2
π
=π1sin
π
22cos< 0,
signsignp p
pp p












(8)
and since

we conclude that 0

 with equality iff
.
=2p
We are now ready to prove the announced double
inequality concerning

,.
M
np
Proposition
For and all positive integers
1p ,n
 
4,
!
nMnp Mn
n

,
2 with left (resp. right) equality
iff (resp. ), whenever . =1p=2p1n
Proof:
Based on Lemma III.1 we can see immediately that for
, with equa-
lity iff
1< <<tp
=2.p
 
,< ,,2,MntMnp Mn
p We only need to examine (2) whenever 1

:
It is clear that

,1= 4nn
M
n
s, with


1
=1
=
1
n
nn
k
k
s
kn1 s
k



. Since and =1
1
=1
==
n
k
n
skkn11
11
s
kkn


 
 

 
n

we see that
!
1
=n sn and we are done.
Remark 1
Independently of the above approach one can easily
verify that n
s
converges for all by the logarithmic
series test along the lines of Remark II. 1(i): evidently
and also for (by the crude logarithmic
inequality)
n
>0
n
s2n
n
log n s


, where


=2 =1
n
nm
mk


=<.
1
nm
n
k
k



n

Thus we obtain the (crude but not obvious) inequality
1
!
n
e
n


 for 2.n
We conclude this work with two closed type formulae
that are automatically true when we set 2p

in (2).
Exploiting the fact that

2
22
4π
,2
2
n
n
Bn
n
 


Mn and
the classical value of 2
n



for even and odd (see n 

e.g.[2]) we arrive, respectively, for any at the
following fomulae:
mN
Corollary 1
 

1
22
21
m
k



21
4
22π
=!
22
m
m
kkm
m






=1 k


Corollary 2
 
 
1
21
22
m
k

2
=1
8
223π
=23!!
m
m
k
kkm
m







21
Remark 2
It is clear that the above two formulae provide us with
the seemingly untabulated formula



 
1
=1
22
212
kk
kk
2223!!
=
2 32!
m
k
m m
m m


 


.
4. References
[1] G. Andrews, R. Askey and R. Roy, “Special Functions,”
Cambridge University Press, Cambridge, 1999.
[2] I. Gradshteyn and I. Ryzhik, “Tables of Integrals, Series
and Products,” Academic Press, Waltham, 1980.
[3] D. Karayannakis, “An algorithm for the Evaluation of the
Gamma function and Ramifications. Part I,” International
Journal of Mathematics, Game Theory and Algebra, Vol.
19, No. 4, 2010.
[4] L. E. Lutwak, “Selected Affine Isoperimetric Inequalities,
Handbook of Convex Geometry,” North-Holland Pub-
lishing Co., Amsterdam, 1993.