Energy integrals, metric embeddings and absolutely
summing operators
Daniel Galicer1
Joint work with Daniel Carando and Damián Pinasco
1 Universidad
de Buenos Aires - CONICET;
University of Warwick - June 2015
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
W. A. WILSON, On certain types of continuous transformations
of metric spaces, Amer. J. of Math., Vol. 57 (1935), pp. 62-68.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
W. A. WILSON, On certain types of continuous transformations
of metric spaces, Amer. J. of Math., Vol. 57 (1935), pp. 62-68.
Wilson investigated certain properties of those metric spaces which
arise from a metric space by taking as its new metric a suitable (one
variable) function of the old one.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
W. A. WILSON, On certain types of continuous transformations
of metric spaces, Amer. J. of Math., Vol. 57 (1935), pp. 62-68.
Wilson investigated certain properties of those metric spaces which
arise from a metric space by taking as its new metric a suitable (one
variable) function of the old one.
(X, d)
f
(X, f (d)), where f (d)(x, y) := f (d(x, y)).
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
For the metric space (R, | · |), Wilson considered the function
f (t) = t1/2 .
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
For the metric space (R, | · |), Wilson considered the function
f (t) = t1/2 .
Denote d1/2 := f (| · |)
d1/2 (x, y) = |x − y|1/2 .
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
For the metric space (R, | · |), Wilson considered the function
f (t) = t1/2 .
Denote d1/2 := f (| · |)
d1/2 (x, y) = |x − y|1/2 .
• He showed that (R, d1/2 ) may be isometrically imbedded in a
separable Hilbert space.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
For the metric space (R, | · |), Wilson considered the function
f (t) = t1/2 .
Denote d1/2 := f (| · |)
d1/2 (x, y) = |x − y|1/2 .
• He showed that (R, d1/2 ) may be isometrically imbedded in a
separable Hilbert space.
In other words, he proved that there exist a distance preserving
mapping (isometry)
j : R, d1/2 → `2 , k · k`2
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
For the metric space (R, | · |), Wilson considered the function
f (t) = t1/2 .
Denote d1/2 := f (| · |)
d1/2 (x, y) = |x − y|1/2 .
• He showed that (R, d1/2 ) may be isometrically imbedded in a
separable Hilbert space.
In other words, he proved that there exist a distance preserving
mapping (isometry)
j : R, d1/2 → `2 , k · k`2
That is,
kj(x) − j(y)k`2 = d1/2 (x, y) = |x − y|1/2 , ∀x, y ∈ R.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
I. Schoenberg and J. von Neumann, Fourier integrals and metric
geometry, Trans. of the Amer. Math. Soc., Vol. 50, No. 2 (1941),
pp. 226-251.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
I. Schoenberg and J. von Neumann, Fourier integrals and metric
geometry, Trans. of the Amer. Math. Soc., Vol. 50, No. 2 (1941),
pp. 226-251.
They characterized
those function f for which the metric space
R, f (| · |) can be isometrically imbedded in a Hilbert space.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
I. Schoenberg and J. von Neumann, Fourier integrals and metric
geometry, Trans. of the Amer. Math. Soc., Vol. 50, No. 2 (1941),
pp. 226-251.
They characterized
those function f for which the metric space
R, f (| · |) can be isometrically imbedded in a Hilbert space.
They proved that, for 0 < α < 1, f (t) = tα becomes a suitable metric
transformation.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
What happens for the n-dimensional real space Rn ?
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
What happens for the n-dimensional real space Rn ?
Is the metric space Rn , dα ) isometrically imbeddable in `2 , where
dα (x, y) = kx − ykα ?
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
What happens for the n-dimensional real space Rn ?
Is the metric space Rn , dα ) isometrically imbeddable in `2 , where
dα (x, y) = kx − ykα ?
I. Schoenberg, On certain metric spaces arising from Euclidean
spaces by a change of metric and their imbedding in Hilbert
space, Ann. of Math. 38 (1937), pp. 787-793.
Theorem (Schoenberg)
For 0 < α < 1, the metric space Rn , dα is imbeddable in `2 .
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Moreover, by combining Schoenberg’s proof and a classic result of
n
Menger,
we have that for every compact set K ⊂ R the metric space
K, dα may be imbeddable in the surface of a Hilbert sphere.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Moreover, by combining Schoenberg’s proof and a classic result of
n
Menger,
we have that for every compact set K ⊂ R the metric space
K, dα may be imbeddable in the surface of a Hilbert sphere.
Classic result
For every compact set K ⊂ Rn , there exist a positive number r and a
distance preserving mapping
j : K, dα → rS`2 , k · k`2
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Moreover, by combining Schoenberg’s proof and a classic result of
n
Menger,
we have that for every compact set K ⊂ R the metric space
K, dα may be imbeddable in the surface of a Hilbert sphere.
Classic result
For every compact set K ⊂ Rn , there exist a positive number r and a
distance preserving mapping
j : K, dα → rS`2 , k · k`2
Note that kj(x)k`2 = r, ∀x ∈ K.
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Moreover, by combining Schoenberg’s proof and a classic result of
n
Menger,
we have that for every compact set K ⊂ R the metric space
K, dα may be imbeddable in the surface of a Hilbert sphere.
Classic result
For every compact set K ⊂ Rn , there exist a positive number r and a
distance preserving mapping
j : K, dα → rS`2 , k · k`2
Note that kj(x)k`2 = r, ∀x ∈ K.
It is natural to define,
ρα (K) := inf r
Classic results
New results
Metric spaces arising from Euclidean spaces by a change of
metric: some history
Moreover, by combining Schoenberg’s proof and a classic result of
n
Menger,
we have that for every compact set K ⊂ R the metric space
K, dα may be imbeddable in the surface of a Hilbert sphere.
Classic result
For every compact set K ⊂ Rn , there exist a positive number r and a
distance preserving mapping
j : K, dα → rS`2 , k · k`2
Note that kj(x)k`2 = r, ∀x ∈ K.
It is natural to define,
ρα (K) := inf r
least possible radius (α-Schoenberg’s radii of K)
Classic results
A connection with another area
All these results can be framed within a vast area called "metric
geometry".
New results
Classic results
New results
A connection with another area
All these results can be framed within a vast area called "metric
geometry".
Link
Metric Geometry ↔ Potential Theory
Classic results
Energy Integrals: some definitions
Let K ⊂ Rn be a compact set and µ be a signed Borel measure
supported on K of total mass one (i.e., µ(K) = 1.)
New results
Classic results
New results
Energy Integrals: some definitions
Let K ⊂ Rn be a compact set and µ be a signed Borel measure
supported on K of total mass one (i.e., µ(K) = 1.)
For a real number p (for us, 0 < p < 2), we define
Z Z
Ip (µ; K) :=
kx−ykp dµ(x)dµ(y)
K
K
Classic results
New results
Energy Integrals: some definitions
Let K ⊂ Rn be a compact set and µ be a signed Borel measure
supported on K of total mass one (i.e., µ(K) = 1.)
For a real number p (for us, 0 < p < 2), we define
Z Z
Ip (µ; K) :=
kx−ykp dµ(x)dµ(y)
p-energy integral given by µ.
K
K
Classic results
New results
Energy Integrals: some definitions
Let K ⊂ Rn be a compact set and µ be a signed Borel measure
supported on K of total mass one (i.e., µ(K) = 1.)
For a real number p (for us, 0 < p < 2), we define
Z Z
Ip (µ; K) :=
kx−ykp dµ(x)dµ(y)
p-energy integral given by µ.
K
K
And define,
Mp (K) := sup Ip (µ; K)
µ
Classic results
New results
Energy Integrals: some definitions
Let K ⊂ Rn be a compact set and µ be a signed Borel measure
supported on K of total mass one (i.e., µ(K) = 1.)
For a real number p (for us, 0 < p < 2), we define
Z Z
Ip (µ; K) :=
kx−ykp dµ(x)dµ(y)
p-energy integral given by µ.
K
K
And define,
Mp (K) := sup Ip (µ; K)
µ
p-maximal energy of K.
Classic results
New results
The connection!
R. Alexander and K.B. Stolarsky, Extremal problems of distance
geometry related to energy integrals. Trans. Am. Math. Soc. 193
(1974), pp. 1-31.
Classic results
New results
The connection!
R. Alexander and K.B. Stolarsky, Extremal problems of distance
geometry related to energy integrals. Trans. Am. Math. Soc. 193
(1974), pp. 1-31.
Theorem (Alexander-Stolarsky)
Let K ⊂ Rn be a compact set. Then,
r
ρα (K) =
M2α (K)
2
Classic results
New results
The connection!
R. Alexander and K.B. Stolarsky, Extremal problems of distance
geometry related to energy integrals. Trans. Am. Math. Soc. 193
(1974), pp. 1-31.
Theorem (Alexander-Stolarsky)
Let K ⊂ Rn be a compact set. Then,
r
ρα (K) =
M2α (K)
2
We will be focused on computing the value of M2α (K).
Classic results
New results
Some results...
Denote by Bn the unit ball in Rn .
M1 (B1 ) = M1 ([−1, 1]) = 1 (Alexander-Stolarsky, Trans. AMS.
’74)
Classic results
New results
Some results...
Denote by Bn the unit ball in Rn .
M1 (B1 ) = M1 ([−1, 1]) = 1 (Alexander-Stolarsky, Trans. AMS.
’74)
M1 (B3 ) = 2 (Alexander, Proc. AMS. ’77)
Classic results
New results
Some results...
Denote by Bn the unit ball in Rn .
M1 (B1 ) = M1 ([−1, 1]) = 1 (Alexander-Stolarsky, Trans. AMS.
’74)
M1 (B3 ) = 2 (Alexander, Proc. AMS. ’77)
M1 (Bn ) =???
Classic results
New results
Some results...
Denote by Bn the unit ball in Rn .
M1 (B1 ) = M1 ([−1, 1]) = 1 (Alexander-Stolarsky, Trans. AMS.
’74)
M1 (B3 ) = 2 (Alexander, Proc. AMS. ’77)
M1 (Bn ) =???
remained unknown for a very long time.
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Question
What is the value of Mp (Bn ), for 0 < p < 2?
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Question
What is the value of Mp (Bn ), for 0 < p < 2?
What is the value of Mp (K) for other type of convex bodies K t’s
(e.g., an ellipsoid or the unit ball of `nq ?)
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Question
What is the value of Mp (Bn ), for 0 < p < 2?
What is the value of Mp (K) for other type of convex bodies K t’s
(e.g., an ellipsoid or the unit ball of `nq ?)
Does the number
π 1/2 Γ( n+1
)
2
Γ( n2 )
look familiar to you?
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Question
What is the value of Mp (Bn ), for 0 < p < 2?
What is the value of Mp (K) for other type of convex bodies K t’s
(e.g., an ellipsoid or the unit ball of `nq ?)
Does the number
π 1/2 Γ( n+1
)
2
Γ( n2 )
look familiar to you?
Classic results
New results
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
Theorem (Hinrichs, Nickolas and Wolf)
M1 (Bn ) =
π 1/2 Γ( n+1
2 )
.
n
Γ( 2 )
Question
What is the value of Mp (Bn ), for 0 < p < 2?
What is the value of Mp (K) for other type of convex bodies K t’s
(e.g., an ellipsoid or the unit ball of `nq ?)
Does the number
The number
π 1/2 Γ( n+1
)
2
Γ( 2n )
π 1/2 Γ( n+1
)
2
Γ( n2 )
look familiar to you?
is exactly π1 (id : `n2 → `n2 ).
Classic results
New results
Theorem (Carando, G., Pinasco: Int Math Res Notices)
π 1/2 Γ n+p
Mp (Bn ) =
2 n Mp ([−1, 1])
Γ p+1
2 Γ 2
Classic results
New results
Theorem (Carando, G., Pinasco: Int Math Res Notices)
π 1/2 Γ n+p
Mp (Bn ) =
2 n Mp ([−1, 1])
Γ p+1
2 Γ 2
= πp (id : `n2 → `n2 )p Mp ([−1, 1])
Classic results
New results
Theorem (Carando, G., Pinasco: Int Math Res Notices)
π 1/2 Γ n+p
Mp (Bn ) =
2 n Mp ([−1, 1])
Γ p+1
2 Γ 2
= πp (id : `n2 → `n2 )p Mp ([−1, 1])
Using limm→∞
Γ(m+c)
Γ(m)mc
= 1, and the previous result we get:
Corollary
α
ρα (Bn ) n 2 .
Classic results
New results
Theorem (Carando, G., Pinasco: Int Math Res Notices)
π 1/2 Γ n+p
Mp (Bn ) =
2 n Mp ([−1, 1])
Γ p+1
2 Γ 2
= πp (id : `n2 → `n2 )p Mp ([−1, 1])
Using limm→∞
Γ(m+c)
Γ(m)mc
= 1, and the previous result we get:
Corollary
α
ρα (Bn ) n 2 .
In the case where the convex set is an ellipsoid E?
Mp (E) = Mp (T(Bn )) = πp (T : `n2 → `n2 )p Mp ([−1, 1])
Classic results
New results
How is the p-summing norm related with this problem?
Classic results
New results
How is the p-summing norm related with this problem?
Lemma
For every x ∈ Rn , we have
p
kTxk = πp (T :
`n2
→
`n2 )p
Z
|hx, ti|p dν(t),
Sn−1
where ν is a probability measure on the unit sphere Sn−1 .
Classic results
New results
How is the p-summing norm related with this problem?
Lemma
For every x ∈ Rn , we have
p
kTxk = πp (T :
`n2
→
`n2 )p
Z
|hx, ti|p dν(t),
Sn−1
where ν is a probability measure on the unit sphere Sn−1 .
If T is the identity...
p
kxk = πp (id :
`n2
→
`n2 )p
Z
|hx, ti|p dλ(t),
Sn−1
where λ is just the normalized Lebesgue surface measure on the
sphere Sn−1 .
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Bn
Bn
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn
Bn
Sn−1
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
Bn
Bn
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
Bn
Bn
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
Bn
Bn
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
= πp (id`n2 )p
Z
Sn−1
Bn Bn
1 Z 1
hZ
−1
−1
i
|u − v|p dµt (u)dµt (v) dλ(t)
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
Z
Bn Bn
1 Z 1
hZ
i
= πp (id`n2 )p
|u − v|p dµt (u)dµt (v) dλ(t)
n−1
−1 −1
ZS
≤ πp (id`n2 )p
Mp ([−1, 1])dλ(t)
Sn−1
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
Upper bound: sketch
Let µ be a signed borel measure on Bn of total mass one.
Z Z
Ip (µ; Bn ) :=
kx − ykp dµ(x)dµ(y)
Z
Z BnZ Bn
p
=
πp (id`n2 )
|hx − y, ti|p dλ(t)dµ(x)dµ(y)
Bn Bn
Sn−1
Z
hZ Z
i
= πp (id`n2 )p
|hx − y, ti|p dµ(x)dµ(y) dλ(t)
Sn−1
Z
Bn Bn
1 Z 1
hZ
i
= πp (id`n2 )p
|u − v|p dµt (u)dµt (v) dλ(t)
n−1
−1 −1
ZS
≤ πp (id`n2 )p
Mp ([−1, 1])dλ(t)
Sn−1
p
= πp (id ) Mp ([−1, 1])
`n2
Classic results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
How to get equality?
New results
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
How to get equality? Recall that for any µ,
Ip (µ; Bn ) = πp (id`n2 )p
Z
Sn−1
hZ
1
−1
Z
1
−1
i
|u − v|p dµt (u)dµt (v) dλ(t)
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
How to get equality? Recall that for any µ,
Ip (µ; Bn ) = πp (id`n2 )p
Z
Sn−1
hZ
1
−1
Z
1
i
|u − v|p dµt (u)dµt (v) dλ(t)
−1
We found a sequence (µk )k∈N of signed measures of total mass one Bn
such that
Z 1Z 1
|u − v|p dµkt (u)dµkt (v) ⇒ Mp ([−1, 1]).
−1
−1
Classic results
New results
Mp (Bn ) = πp (id`n2 )p Mp ([−1, 1])
How to get equality? Recall that for any µ,
Ip (µ; Bn ) = πp (id`n2 )p
Z
Sn−1
hZ
1
−1
Z
1
i
|u − v|p dµt (u)dµt (v) dλ(t)
−1
We found a sequence (µk )k∈N of signed measures of total mass one Bn
such that
Z 1Z 1
|u − v|p dµkt (u)dµkt (v) ⇒ Mp ([−1, 1]).
−1
−1
Therefore, Mp (Bn ) ≥ Ip (µk ; Bn ) → πp (id`n2 )p Mp ([−1, 1]).
Classic results
New results
Bounds for other convex bodies
Let K ⊂ Rn be a centrally symmetric convex body, then
K is just the
unit ball of an n-dimensional Banach space E, k · kE
Classic results
New results
Bounds for other convex bodies
Let K ⊂ Rn be a centrally symmetric convex body, then
K is just the
unit ball of an n-dimensional Banach space E, k · kE
i.e., K = BE .
Classic results
New results
Bounds for other convex bodies
Let K ⊂ Rn be a centrally symmetric convex body, then
K is just the
unit ball of an n-dimensional Banach space E, k · kE
i.e., K = BE .
Question
How can we estimate the value of ρα (BE ), 0 < α < 1?
Classic results
New results
Bounds for other convex bodies
Let K ⊂ Rn be a centrally symmetric convex body, then
K is just the
unit ball of an n-dimensional Banach space E, k · kE
i.e., K = BE .
Question
How can we estimate the value of ρα (BE ), 0 < α < 1? Or,
equivalently, how can we compute Mp (BE ), 0 < p < 2?
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
This bound is expressed in terms of the mean width of BE , and is good
enough in many cases!
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
This bound is expressed in terms of the mean width of BE , and is good
enough in many cases!
Theorem (Carando, G., Pinasco)
Let 1 < q ≤ 2 then
p
Mp (B`nq ) n q0 .
α
In particular, ρα (B`nq ) n q0 .
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
This bound is expressed in terms of the mean width of BE , and is good
enough in many cases!
Theorem (Carando, G., Pinasco)
Let 1 < q ≤ 2 then
p
Mp (B`nq ) n q0 .
α
In particular, ρα (B`nq ) n q0 .
Remark:
Upper bounds were given using the previous result.
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
This bound is expressed in terms of the mean width of BE , and is good
enough in many cases!
Theorem (Carando, G., Pinasco)
Let 1 < q ≤ 2 then
p
Mp (B`nq ) n q0 .
α
In particular, ρα (B`nq ) n q0 .
Remark:
Upper bounds were given using the previous result.
For the lower bound we use n(q−2)/2q B`n2 ⊂ B`nq and the fact that
Mp (·) is monotone function (under inclusion).
Classic results
New results
Theorem
(General upper bound)
Z
π 1/2 Γ n+p
2
ktkpE0 dλ(t).
Mp (BE ) ≤ Mp ([−1, 1]) p+1 n n−1
Γ 2 Γ 2 S
This bound is expressed in terms of the mean width of BE , and is good
enough in many cases!
Theorem (Carando, G., Pinasco)
Let 1 < q ≤ 2 then
p
Mp (B`nq ) n q0 .
α
In particular, ρα (B`nq ) n q0 .
Remark:
Upper bounds were given using the previous result.
For the lower bound we use n(q−2)/2q B`n2 ⊂ B`nq and the fact that
Mp (·) is monotone function (under inclusion).
Classic results
New results
Several open questions
What is the asymptotic behavior of ρα (B`nq ), for 2 ≤ q ≤ ∞ or
q = 1?
Classic results
New results
Several open questions
What is the asymptotic behavior of ρα (B`nq ), for 2 ≤ q ≤ ∞ or
q = 1?
Is there a closed formula for Mp ([−1, 1])?
Classic results
New results
R. Alexander, K.B. Stolarsky, Extremal problems of distance
geometry related to energy integrals. Trans. Am. Math. Soc. 193
(1974), pp. 1-31.
D. Carando, D. Galicer, D. Pinasco, Energy Integrals and Metric
Embedding Theory, Int Math Res Notices, to appear.
A. Hinrichs, P. Nickolas, R. Wolf, A note on the metric geometry
of the unit ball, Math. Z. 268 (2011), pp. 887-896.
I. Schoenberg, On certain metric spaces arising from Euclidean
spaces by a change of metric and their imbedding in Hilbert
space, Ann. of Math. 38 (1937), pp. 787-793.
J. von Neumann and I. Schoenberg, Fourier integrals and metric
geometry, Trans. of the Amer. Math. Soc., Vol. 50, No. 2 (1941),
pp. 226-251.
W. A. Wilson, On certain types of continuous transformations of
metric spaces, Amer. J. of Math., Vol. 57 (1935), pp. 62-68.