Download A sharp Rogers-Shephard type inequality for the p

Chiara Bianchini & Andrea Colesanti
Università degli Studi di Firenze
A sharp Rogers-Shephard type inequality for
the p-difference body of a planar convex body
“Affine Invariants, Randomness, and Approximation
in Convex Geometry”
Hoboken, April 14-15, 2007
The difference body of a convex body
The difference body of a convex body
◮
Let K be a convex body in Rn ; the difference body of K is
DK := K + (−K ) = {x + y | x ∈ K , −y ∈ K } .
DK is a centrally symmetric convex body.
The difference body of a convex body
◮
Let K be a convex body in Rn ; the difference body of K is
DK := K + (−K ) = {x + y | x ∈ K , −y ∈ K } .
DK is a centrally symmetric convex body.
◮
A straightforward application of the Brunn-Minkowski
inequality gives
Vn (DK ) ≥ 2n Vn (K )
for every convex body K (here Vn denotes the n-dimensional
volume).
The Rogers-Shephard inequality
The Rogers-Shephard inequality
Theorem (Rogers & Shephard, 1957). For every convex body K
2n
Vn (DK ) ≤
Vn (K ) .
n
The Rogers-Shephard inequality
Theorem (Rogers & Shephard, 1957). For every convex body K
2n
Vn (DK ) ≤
Vn (K ) .
n
Equality holds if and only if K is a simplex.
The Rogers-Shephard inequality
Theorem (Rogers & Shephard, 1957). For every convex body K
2n
Vn (DK ) ≤
Vn (K ) .
n
Equality holds if and only if K is a simplex.
Theorem (Rogers & Shephard, 1958). Let K be a convex body
containing the origin and let conv(K ∪ (−K )) be the convex hull of
K ∪ (−K ).
The Rogers-Shephard inequality
Theorem (Rogers & Shephard, 1957). For every convex body K
2n
Vn (DK ) ≤
Vn (K ) .
n
Equality holds if and only if K is a simplex.
Theorem (Rogers & Shephard, 1958). Let K be a convex body
containing the origin and let conv(K ∪ (−K )) be the convex hull of
K ∪ (−K ).Then
Vn (conv(K ∪ (−K ))) ≤ 2n Vn (K ) .
The Rogers-Shephard inequality
Theorem (Rogers & Shephard, 1957). For every convex body K
2n
Vn (DK ) ≤
Vn (K ) .
n
Equality holds if and only if K is a simplex.
Theorem (Rogers & Shephard, 1958). Let K be a convex body
containing the origin and let conv(K ∪ (−K )) be the convex hull of
K ∪ (−K ).Then
Vn (conv(K ∪ (−K ))) ≤ 2n Vn (K ) .
Equality holds if and only if K is a simplex with one vertex at the
origin.
The p-sum of convex bodies (Firey, 1962)
The p-sum of convex bodies (Firey, 1962)
Let p ≥ 1. Let K and L be convex bodies in Rn , containing the
origin, and denote by hK and hL the corresponding support
functions.
The p-sum of convex bodies (Firey, 1962)
Let p ≥ 1. Let K and L be convex bodies in Rn , containing the
origin, and denote by hK and hL the corresponding support
functions.
The p-sum of K and L
K +p L
is the convex body having as support function
p
hK +p L = (hK
+ hLp )1/p .
The p-sum of convex bodies (Firey, 1962)
Let p ≥ 1. Let K and L be convex bodies in Rn , containing the
origin, and denote by hK and hL the corresponding support
functions.
The p-sum of K and L
K +p L
is the convex body having as support function
p
hK +p L = (hK
+ hLp )1/p .
For p = ∞ set
hK +∞ L = max{hK , hL } .
The p-sum of convex bodies (Firey, 1962)
Let p ≥ 1. Let K and L be convex bodies in Rn , containing the
origin, and denote by hK and hL the corresponding support
functions.
The p-sum of K and L
K +p L
is the convex body having as support function
p
hK +p L = (hK
+ hLp )1/p .
For p = ∞ set
hK +∞ L = max{hK , hL } .
Note that
K +1 L = K + L and
K +∞ L = conv(K ∪ L) .
The p-difference body
The p-difference body
Let K be a convex body in Rn containing the origin and let
p ∈ [1, ∞].
The p-difference body
Let K be a convex body in Rn containing the origin and let
p ∈ [1, ∞]. The p-difference body of K is
Dp K = K +p (−K ) .
The p-difference body
Let K be a convex body in Rn containing the origin and let
p ∈ [1, ∞]. The p-difference body of K is
Dp K = K +p (−K ) .
◮
For p = 1: D1 K = DK and
Vn (D1 K ) ≤
2n
Vn (K ) .
n
The p-difference body
Let K be a convex body in Rn containing the origin and let
p ∈ [1, ∞]. The p-difference body of K is
Dp K = K +p (−K ) .
◮
For p = 1: D1 K = DK and
Vn (D1 K ) ≤
◮
2n
Vn (K ) .
n
For p = ∞: D∞ K = conv(K ∪ (−K )) and
Vn (D∞ K ) ≤ 2n Vn (K ) .
The problem
The problem
Given p > 1, find the optimal constant c = c(n, p) such that for
every convex body K in Rn containing the origin
Vn (Dp K ) ≤ c(n, p)Vn (K ) .
The problem
Given p > 1, find the optimal constant c = c(n, p) such that for
every convex body K in Rn containing the origin
Vn (Dp K ) ≤ c(n, p)Vn (K ) .
Remark. For every p and K
Dp K ⊂ DK
and Dp K ⊂ 21/p D∞ K .
The problem
Given p > 1, find the optimal constant c = c(n, p) such that for
every convex body K in Rn containing the origin
Vn (Dp K ) ≤ c(n, p)Vn (K ) .
Remark. For every p and K
Dp K ⊂ DK
and Dp K ⊂ 21/p D∞ K .
Consequently
Vn (Dp K ) ≤ min
n(1+p)
2n
p
,2
Vn (K ) .
n
The two-dimensional case
The two-dimensional case
Theorem (C. Bianchini & A. C., 2006). For every planar
convex body K containing the origin and for every p > 1 we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) ,
where
The two-dimensional case
Theorem (C. Bianchini & A. C., 2006). For every planar
convex body K containing the origin and for every p > 1 we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) ,
where
c(2, p) = 2 + 2(p − 1)
Z
π
2
0
sinp−2 t cosp−2 t
2 (p−1) dt .
p
p
sin t + cosp t
The two-dimensional case
Theorem (C. Bianchini & A. C., 2006). For every planar
convex body K containing the origin and for every p > 1 we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) ,
where
c(2, p) = 2 + 2(p − 1)
Z
π
2
0
sinp−2 t cosp−2 t
2 (p−1) dt .
p
p
sin t + cosp t
Equality holds if K is a triangle with one vertex at the origin.
The two-dimensional case
Theorem (C. Bianchini & A. C., 2006). For every planar
convex body K containing the origin and for every p > 1 we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) ,
where
c(2, p) = 2 + 2(p − 1)
Z
π
2
0
sinp−2 t cosp−2 t
2 (p−1) dt .
p
p
sin t + cosp t
Equality holds if K is a triangle with one vertex at the origin.
Linear parameter systems
Linear parameter systems
A linear parameter system is a family of convex bodies {Kt } that
can be written in the form
Kt = conv{xi + λi tv : i ∈ I },
t ∈ [a, b] ;
Linear parameter systems
A linear parameter system is a family of convex bodies {Kt } that
can be written in the form
Kt = conv{xi + λi tv : i ∈ I },
where: I is an arbitrary index set,
t ∈ [a, b] ;
Linear parameter systems
A linear parameter system is a family of convex bodies {Kt } that
can be written in the form
Kt = conv{xi + λi tv : i ∈ I },
t ∈ [a, b] ;
where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded
subsets of Rn and of R respectively
Linear parameter systems
A linear parameter system is a family of convex bodies {Kt } that
can be written in the form
Kt = conv{xi + λi tv : i ∈ I },
t ∈ [a, b] ;
where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded
subsets of Rn and of R respectively and v ∈ Rn is the direction of
the linear parameter system.
Linear parameter systems
A linear parameter system is a family of convex bodies {Kt } that
can be written in the form
Kt = conv{xi + λi tv : i ∈ I },
t ∈ [a, b] ;
where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded
subsets of Rn and of R respectively and v ∈ Rn is the direction of
the linear parameter system.
Kt
Kt
0
v
1
Kt
2
Convexity of the volume
Convexity of the volume
Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a
linear parameter system is a convex function of t.
Convexity of the volume
Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a
linear parameter system is a convex function of t.
◮
Campi, C. & Gronchi, 1999 (Sylvester’s problem).
Convexity of the volume
Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a
linear parameter system is a convex function of t.
◮
Campi, C. & Gronchi, 1999 (Sylvester’s problem).
◮
Campi & Gronchi, 2002 (Lp -centroid inequality).
Convexity of the volume
Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a
linear parameter system is a convex function of t.
◮
Campi, C. & Gronchi, 1999 (Sylvester’s problem).
◮
Campi & Gronchi, 2002 (Lp -centroid inequality).
◮
Campi & Gronchi, 2006 (volume product).
Convexity of the volume
Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a
linear parameter system is a convex function of t.
◮
Campi, C. & Gronchi, 1999 (Sylvester’s problem).
◮
Campi & Gronchi, 2002 (Lp -centroid inequality).
◮
Campi & Gronchi, 2006 (volume product).
◮
Meyer & Reisner, 2006 (volume product).
p-sums of linear parameter systems
p-sums of linear parameter systems
Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and
{Lt }t∈[a,b] be linear parameter systems along the same direction
s.t. 0 ∈ Kt , Lt for every t, then
{Kt +p Lt }t∈[a,b]
is also a linear parameter system.
p-sums of linear parameter systems
Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and
{Lt }t∈[a,b] be linear parameter systems along the same direction
s.t. 0 ∈ Kt , Lt for every t, then
{Kt +p Lt }t∈[a,b]
is also a linear parameter system.
p = 1: Campi & Gronchi, 2006.
p-sums of linear parameter systems
Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and
{Lt }t∈[a,b] be linear parameter systems along the same direction
s.t. 0 ∈ Kt , Lt for every t, then
{Kt +p Lt }t∈[a,b]
is also a linear parameter system.
p = 1: Campi & Gronchi, 2006.
Corollary. If {Kt }t∈[a,b] is a linear parameter system s.t. 0 ∈ Kt
for every t, then
{Dp Kt }t∈[a,b]
is also a linear parameter system.
p-sums of linear parameter systems
Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and
{Lt }t∈[a,b] be linear parameter systems along the same direction
s.t. 0 ∈ Kt , Lt for every t, then
{Kt +p Lt }t∈[a,b]
is also a linear parameter system.
p = 1: Campi & Gronchi, 2006.
Corollary. If {Kt }t∈[a,b] is a linear parameter system s.t. 0 ∈ Kt
for every t, then
{Dp Kt }t∈[a,b]
is also a linear parameter system. In particular
Vn (Dp Kt ) is a convex function of t.
Parallel chord movements
Parallel chord movements
These are particular linear parameter systems:
Kt = {x + α(x)tv : x ∈ K },
t ∈ [a, b],
where the functions α is constant on each chord parallel to v and
such that convexity is preserved for every t.
Parallel chord movements
These are particular linear parameter systems:
Kt = {x + α(x)tv : x ∈ K },
t ∈ [a, b],
where the functions α is constant on each chord parallel to v and
such that convexity is preserved for every t.
K
Kt
v
Parallel chord movements
These are particular linear parameter systems:
Kt = {x + α(x)tv : x ∈ K },
t ∈ [a, b],
where the functions α is constant on each chord parallel to v and
such that convexity is preserved for every t.
K
Kt
v
Remark. The volume is constant along a parallel chord movement.
The first main ingredient of the proof
The first main ingredient of the proof
Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex
bodies s.t. 0 ∈ Kt for every t, then
Vn (Dp Kt )
,
Vn (Kt )
is a convex function of t.
t ∈ [a, b] ,
The first main ingredient of the proof
Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex
bodies s.t. 0 ∈ Kt for every t, then
Vn (Dp Kt )
,
Vn (Kt )
t ∈ [a, b] ,
is a convex function of t. In particular the above quotient attains
its maximum either for t = a or for t = b.
The first main ingredient of the proof
Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex
bodies s.t. 0 ∈ Kt for every t, then
Vn (Dp Kt )
,
Vn (Kt )
t ∈ [a, b] ,
is a convex function of t. In particular the above quotient attains
its maximum either for t = a or for t = b.
Proof.
◮
{Dp Kt }t∈[a,b] is a linear parameter system and then Vn (Dp Kt )
is convex w.r.t. t.
The first main ingredient of the proof
Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex
bodies s.t. 0 ∈ Kt for every t, then
Vn (Dp Kt )
,
Vn (Kt )
t ∈ [a, b] ,
is a convex function of t. In particular the above quotient attains
its maximum either for t = a or for t = b.
Proof.
◮
{Dp Kt }t∈[a,b] is a linear parameter system and then Vn (Dp Kt )
is convex w.r.t. t.
◮
Vn (Kt ) is constant w.r.t. t.
Polygons
Polygons
For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P,
there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for
every t and
◮
P ∈ {Pt }t∈[a,b] ;
Polygons
For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P,
there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for
every t and
◮
P ∈ {Pt }t∈[a,b] ;
◮
Pa and Pb are both polygons having at most (k − 1) vertices.
Polygons
For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P,
there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for
every t and
◮
P ∈ {Pt }t∈[a,b] ;
◮
Pa and Pb are both polygons having at most (k − 1) vertices.
v
v
P
Polygons
For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P,
there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for
every t and
◮
P ∈ {Pt }t∈[a,b] ;
◮
Pa and Pb are both polygons having at most (k − 1) vertices.
v
v
P
V2 (Dp P)
≤ max
V2 (P)
V2 (Dp Pa ) V2 (Dp Pb )
,
V2 (Pa )
V2 (Pb )
From polygons to triangles
From polygons to triangles
◮
For every convex polygon P with k vertices, k > 3, and s.t.
0 ∈ P, there exists a polygon P ′ with at most (k − 1) vertices
and s.t. 0 ∈ P ′ and
V2 (Dp P)
V2 (Dp P ′ )
≤
.
V2 (P)
V2 (P ′ )
From polygons to triangles
◮
For every convex polygon P with k vertices, k > 3, and s.t.
0 ∈ P, there exists a polygon P ′ with at most (k − 1) vertices
and s.t. 0 ∈ P ′ and
V2 (Dp P)
V2 (Dp P ′ )
≤
.
V2 (P)
V2 (P ′ )
◮
For every convex polygon P s.t. 0 ∈ P there exists a triangle
T s.t. 0 ∈ T and
V2 (Dp T )
V2 (Dp P)
≤
.
V2 (P)
V2 (T )
Putting a vertex in the origin in two moves
Putting a vertex in the origin in two moves
Let T ′ be a triangle containing 0. There exists a triangle T with
one vertex at the origin such that
V2 (Dp T ′ )
V2 (Dp T )
≤
.
V2 (T ′ )
V2 (T )
Putting a vertex in the origin in two moves
Let T ′ be a triangle containing 0. There exists a triangle T with
one vertex at the origin such that
V2 (Dp T ′ )
V2 (Dp T )
≤
.
V2 (T ′ )
V2 (T )
To prove it, we use the fact that translations are parallel chord
movements.
Putting a vertex in the origin in two moves
Let T ′ be a triangle containing 0. There exists a triangle T with
one vertex at the origin such that
V2 (Dp T ′ )
V2 (Dp T )
≤
.
V2 (T ′ )
V2 (T )
To prove it, we use the fact that translations are parallel chord
movements.
T’
v
o
o
v
o
Putting a vertex in the origin in two moves
Let T ′ be a triangle containing 0. There exists a triangle T with
one vertex at the origin such that
V2 (Dp T ′ )
V2 (Dp T )
≤
.
V2 (T ′ )
V2 (T )
To prove it, we use the fact that translations are parallel chord
movements.
T’
v
o
v
o
o
v
o
v
o
o
T’
The computation of c(2, p)
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
c(2, p) = sup
V2 (Dp K )
: K = planar convex body, 0 ∈ K
V2 (K )
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
V2 (Dp K )
: K = planar convex body, 0 ∈ K
V2 (K )
V2 (Dp P)
: P = polygon, 0 ∈ T
= sup
V2 (P)
c(2, p) = sup
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
V2 (Dp K )
: K = planar convex body, 0 ∈ K
V2 (K )
V2 (Dp P)
: P = polygon, 0 ∈ T
= sup
V2 (P)
V2 (Dp T )
: T = triangle, 0 ∈ T
= sup
V2 (T )
c(2, p) = sup
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
V2 (Dp K )
V2 (K )
V2 (Dp P)
= sup
V2 (P)
V2 (Dp T )
= sup
V2 (T )
V2 (Dp T )
= sup
V2 (T )
c(2, p) = sup
: K = planar convex body, 0 ∈ K
: P = polygon, 0 ∈ T
: T = triangle, 0 ∈ T
: T = triangle, 0 is a vertex of T
The computation of c(2, p)
c(2, p) is the optimal constant such that for every planar convex
body K containing the origin we have
V2 (Dp K ) ≤ c(2, p)V2 (K ) .
=
=
=
=
V2 (Dp K )
V2 (K )
V2 (Dp P)
sup
V2 (P)
V2 (Dp T )
sup
V2 (T )
V2 (Dp T )
sup
V2 (T )
V2 (Dp T )
, T
V2 (T )
c(2, p) = sup
: K = planar convex body, 0 ∈ K
: P = polygon, 0 ∈ T
: T = triangle, 0 ∈ T
: T = triangle, 0 is a vertex of T
= any triangle with a vertex at 0.