Some remarks on the girth
functions of convex bodies
Paolo Gronchi
Dipartimento di Matematica e Applicazioni per l’Architettura
Università degli Studi di Firenze
(joint work with Stefano Campi)
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1
What is the girth function?
u
u
K
 n 1


ggKK(u)
width(K|u
(K|u) /) 
(u)==mean
perimeter
2
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First property of the girth function
gK (u) is a support function (when extended homogeneously)
unit segment
ball
[u,-u]
n
g K u   V ( K , 
B,
B
,
,
B
,
u
)


2
n2
1
1
  hu z  dS1 K ; z    u, z dS1 K ; z 
2 S n1
2 S n1
area measure
of order 1
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convex
3
gK (u) is the support function of Π1K ,
the projection body of order 1 of K
1
h 1K (u )   u, z dS1 K ; z 
2 S n1
Π1K is a zonoid
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Not all zonoids are
projection bodies of order 1.
All Minkowski
h  K (usums
)  0of disks are
1
projections bodies of order 1.
K = segment parallel to u
Π1K = disk in u
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This suggests that hΠ K (u) is not independent of
1
its values on u.
In dimension 3, such an ellipsoid
cannot be the sum of disks.
In higher dimensions
n for any K,
It is easy to prove that,
1
h 1K (ei ) 
h 1 K ( e j )

h 1K (e1)  h n1K1(e2 )  h 1K (e3 )
j 1
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This suggests that hΠ K (u) is not independent of
1
its values on u.
Integrating last inequality we find
1
h  K u  
4 u
1

 h  K z  dz
1

S2
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h 1K (e1)  h 1K (e2 )  h 1K (e3 )
is equivalent to the existence of a box P such that
h 1P(ei )  h 1K (ei ) , i  1,2,3.
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The existence of a cylinder C such that
u
h 1C (u)  h 1K (u) ,

h 1C (v)  h 1K (v) , v  u ,
is equivalent to the statements
1
h 1K u  
h 1K z  dz

4 u S 2
negative number!

1
h 1K v   h 1K u  
h 1K z  dz

2
4 u S 2
as a function of vu is a support function.
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h 1K u1   h 1K u2   h 1K u1  u2   0

42  u1  u2
 u1  u2

hK u1  u2 
u1  u2 hK u1  u2   hK u1  u2 
2
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n V  K , B ,  , B, u   V  K , B ,  , B, u  
1
2
2
 V K , B,, B, u1  u2  
 V K , B,, B, u1  u2   V K , B,, B, u1  u2 

 V K , B,, B, z , u1 Bu2 VzKdz, B,, B, z , u1  u2  dz
S n1
0
S n1
K , B,, B, u1  u2 , u1  u2
 V
 0
 K  u1, u2
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