Mean-width and mean-norm of isotropic convex bodies Aleksander Pe - - PowerPoint PPT Presentation

Mean-width and mean-norm of isotropic convex bodies

Aleksander Pe lczy´ nski Memorial Conference

Apostolos Giannopoulos

July 16, 2014

(Bedlewo 2014) M and M∗ estimates July 16, 2014 1 / 36

M and M∗

We assume that K is a centrally symmetric convex body of volume 1 in Rn: K = {x ∈ Rn : x 1}. The mean-norm of K is defined by M(K) =

• Sn−1 x dσ(x).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 2 / 36

M and M∗

We assume that K is a centrally symmetric convex body of volume 1 in Rn: K = {x ∈ Rn : x 1}. The mean-norm of K is defined by M(K) =

• Sn−1 x dσ(x).

The support function of K is hK(x) = x∗ = max{x, y : y ∈ K}, and the mean-width of K is M∗(K) = w(K) =

• Sn−1 hK(x) dσ(x).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 2 / 36

Lower bounds

Using integration in polar coordinates and H¨

• lder’s inequality we get

M(K) |Bn

2 |

|K| 1/n c1 √n. From Urysohn’s inequality, M∗(K) vrad(K) := |K| |Bn

2 |

1/n c2 √n.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 3 / 36

Lower bounds

Using integration in polar coordinates and H¨

• lder’s inequality we get

M(K) |Bn

2 |

|K| 1/n c1 √n. From Urysohn’s inequality, M∗(K) vrad(K) := |K| |Bn

2 |

1/n c2 √n. These lower bounds for M and M∗ are sharp: if Dn = rnBn

2 has

volume 1 then rn ≃ √n and M(Dn) = 1 rn ≃ 1 √n while M∗(Dn) = rn ≃ √n.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 3 / 36

Upper bounds

Theorem (Lewis, Figiel-Tomczak, Pisier)

Every centrally symmetric convex body K in Rn has a linear image (a position) ˜ K of volume 1 such that M( ˜ K) M∗( ˜ K) c1 log[d(XK, ℓn

2) + 1] c2 log n.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 4 / 36

Upper bounds

Theorem (Lewis, Figiel-Tomczak, Pisier)

Every centrally symmetric convex body K in Rn has a linear image (a position) ˜ K of volume 1 such that M( ˜ K) M∗( ˜ K) c1 log[d(XK, ℓn

2) + 1] c2 log n.

For this position of K, using the previous lower bounds, we have M( ˜ K) c log n √n and M∗( ˜ K) c√n log n. Question: What can we say about the isotropic position?

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Isotropic convex bodies

Isotropic convex bodies

A convex body K in Rn is called isotropic if it has volume 1, it is centered, and there exists a constant LK > 0 such that

• K

x, θ2dx = L2

K

for every θ ∈ Sn−1.

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Isotropic convex bodies

Isotropic convex bodies

A convex body K in Rn is called isotropic if it has volume 1, it is centered, and there exists a constant LK > 0 such that

• K

x, θ2dx = L2

K

for every θ ∈ Sn−1.

Hyperplane conjecture

There exists an absolute constant C > 0 such that LK C for every n and every isotropic convex body K in Rn. Bourgain: LK c 4 √n log n, Klartag: LK c 4 √n.

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Log-concave measures

Log-concave measures

A measure µ on Rn is called log-concave if µ(λA + (1 − λ)B) µ(A)λµ(B)1−λ for any non-empty compact subsets A and B of Rn and any λ ∈ (0, 1).

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Log-concave measures

Log-concave measures

A measure µ on Rn is called log-concave if µ(λA + (1 − λ)B) µ(A)λµ(B)1−λ for any non-empty compact subsets A and B of Rn and any λ ∈ (0, 1).

Isotropic log-concave measures

We say that a log-concave probability measure µ is isotropic if bar(µ) = 0 and Cov(µ) is the identity matrix:

• xixjfµ(x)dx = δij.

Then, the isotropic constant of µ is Lµ = fµ1/n

∞ ≃ fµ(0)1/n.

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Isotropic log-concave measures

If K is a convex body in Rn, then the Brunn-Minkowski inequality implies that 1K is the density of a log-concave measure. K is isotropic if and only if the measure µK with density Ln

K 1 1

LK K is isotropic. (Bedlewo 2014) M and M∗ estimates July 16, 2014 7 / 36

Isotropic log-concave measures

If K is a convex body in Rn, then the Brunn-Minkowski inequality implies that 1K is the density of a log-concave measure. K is isotropic if and only if the measure µK with density Ln

K 1 1

LK K is isotropic.

Marginal

The marginal of µ with respect to F ∈ Gn,k is defined by πFµ(A) := µ(P−1

F (A)) = µ(A + F ⊥)

for all Borel subsets of F. The density of πFµ is the function fπF µ(x) =

• x+F ⊥ fµ(y)dy,

x ∈ F. If µ is centered, log-concave or isotropic, then πFµ is respectively also centered, log-concave or isotropic.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 7 / 36

Lq-centroid bodies

Lq-centroid bodies

If µ is a probability measure on Rn, the Lq-centroid body Zq(µ), q 1, is the symmetric convex body with support function hZq(µ)(y) := ·, yLq(µ) =

• |x, y|qdµ(x)

1/q .

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Lq-centroid bodies

Lq-centroid bodies

If µ is a probability measure on Rn, the Lq-centroid body Zq(µ), q 1, is the symmetric convex body with support function hZq(µ)(y) := ·, yLq(µ) =

• |x, y|qdµ(x)

1/q . µ is isotropic if and only if it is centered and Z2(µ) = Bn

2 .

From H¨

• lder’s inequality it follows that Z2(µ) ⊆ Zp(µ) ⊆ Zq(µ) for

all 2 p q < ∞. From Borell’s lemma, Zq(µ) ⊆ c q

pZp(µ) for all 2 p < q.

If µ is isotropic, then R(Zq(µ)) := max{hZq(µ)(θ) : θ ∈ Sn−1} cq.

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Lq-centroid bodies

Lq-centroid bodies

If K is a convex body of volume 1 in Rn, the Lq-centroid body Zq(K), q 1, is the symmetric convex body with support function hZq(K)(y) :=

• K

|x, y|qdx 1/q . K is isotropic if and only if it is centered and Z2(K) = LKBn

2 .

If K is centrally symmetric then cK ⊆ Zq(K) ⊆ K for all q n. If K is isotropic and if µK is the isotropic measure with density Ln

K 1 1

LK K, then

Zq(K) = LK Zq(µK).

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The two questions

Assume that K is centrally symmetric and isotropic in Rn.

Question 1

To give an upper bound for M∗(K). From the inclusion K ⊆ (n + 1)LK Bn

2 , one has the obvious bound

M∗(K) (n + 1)LK. Until recently, it was known that M∗(K) cn3/4LK. Several approaches: Hartzoulaki, Pivovarov, “Zq-bound”.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 10 / 36

The two questions

Assume that K is centrally symmetric and isotropic in Rn.

Question 1

To give an upper bound for M∗(K). From the inclusion K ⊆ (n + 1)LK Bn

2 , one has the obvious bound

M∗(K) (n + 1)LK. Until recently, it was known that M∗(K) cn3/4LK. Several approaches: Hartzoulaki, Pivovarov, “Zq-bound”.

Question 2

To give an upper bound for M(K). From the inclusion K ⊇ LK Bn

2 , one has the obvious bound M(K) 1/LK.

Until recently, there was no lower bound depending on n.

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A first idea: Dudley’s entropy estimate

Entropy numbers

The covering number N(K, T) of K by T is the minimal number of translates of T whose union covers K. For any k 1 we set ek(K, T) := inf{s > 0 : N(K, sT) 2k}. The k-th entropy number of K is ek(K) := ek(K, Bn

2 ).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 11 / 36

A first idea: Dudley’s entropy estimate

Entropy numbers

The covering number N(K, T) of K by T is the minimal number of translates of T whose union covers K. For any k 1 we set ek(K, T) := inf{s > 0 : N(K, sT) 2k}. The k-th entropy number of K is ek(K) := ek(K, Bn

2 ).

Dudley’s bound

If K is a centrally symmetric convex body in Rn then √nM∗(K) c1

• k1

1 √ k ek(K, Bn

2 ).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 11 / 36

A first idea: Dudley’s entropy estimate

Covering numbers

If K is an isotropic convex body in Rn then log N(K, sBn

2 ) C1

n3/2LK s for all s > 0. Therefore, ek(K, Bn

2 ) = inf{s > 0 : N(K, sBn 2 ) 2k} C2

√nLK n k .

(Bedlewo 2014) M and M∗ estimates July 16, 2014 12 / 36

A first idea: Dudley’s entropy estimate

Covering numbers

If K is an isotropic convex body in Rn then log N(K, sBn

2 ) C1

n3/2LK s for all s > 0. Therefore, ek(K, Bn

2 ) = inf{s > 0 : N(K, sBn 2 ) 2k} C2

√nLK n k . Then, we combine this with Dudley’s bound M∗(K) c1

• k1

1 √ k ek(K, Bn

2 )

to get M∗(K) Cn3/4LK.

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A refinement by V. Milman and Pisier

The parameter vk(K)

For any k 1 we set vk(K) := sup{vrad(PF(K)) : F ∈ Gn,k}.

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A refinement by V. Milman and Pisier

The parameter vk(K)

For any k 1 we set vk(K) := sup{vrad(PF(K)) : F ∈ Gn,k}.

vk and ek

For every F ∈ Gn,k we have |PF(K)| N(PF(K), ekPF(Bn

2 ))|ekBF|

N(K, ek(K)Bn

2 )ek k |BF| (2ek)k|BF|,

therefore vk(K) 2ek(K).

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A refinement by V. Milman and Pisier

Theorem (V. Milman-Pisier)

For every centrally symmetric convex body K in Rn one has √nM∗(K) c2

n

• k=1

1 √ k Radk(K)vk(K).

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A refinement by V. Milman and Pisier

Theorem (V. Milman-Pisier)

For every centrally symmetric convex body K in Rn one has √nM∗(K) c2

n

• k=1

1 √ k Radk(K)vk(K).

Radk(K)

In the statement above, Radk(K) := sup{Rad(XPF (K)) : F ∈ Gn,k} where Rad(Y ) c3 log(d(Y , ℓdim(Y )

2

) + 1) is the Rademacher constant of Y .

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A refinement by V. Milman and Pisier

Theorem (V. Milman-Pisier)

For every centrally symmetric convex body K in Rn one has √nM∗(K) c2

n

• k=1

1 √ k Radk(K)vk(K).

Radk(K)

In the statement above, Radk(K) := sup{Rad(XPF (K)) : F ∈ Gn,k} where Rad(Y ) c3 log(d(Y , ℓdim(Y )

2

) + 1) is the Rademacher constant of Y . So, roughly speaking, √nM∗(K)

n

• k=1

1 √ k vk(K).

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Mean width: isotropic case

We know that if K is a centrally symmetric isotropic convex body in Rn then Zn(K) ≃ K.

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Mean width: isotropic case

We know that if K is a centrally symmetric isotropic convex body in Rn then Zn(K) ≃ K. So, we look for an upper bound for M∗(Zn(K)), and more generally for M∗(Zq(K)). To this end we use √nM∗(Zq(K))

n

• k=1

1 √ k vk(Zq(K)).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 15 / 36

Mean width: isotropic case

We know that if K is a centrally symmetric isotropic convex body in Rn then Zn(K) ≃ K. So, we look for an upper bound for M∗(Zn(K)), and more generally for M∗(Zq(K)). To this end we use √nM∗(Zq(K))

n

• k=1

1 √ k vk(Zq(K)). In order to estimate vk(Zq(K)) we consider any F ∈ Gn,k and try to give an upper bound for vrad(PF(Zq(K))) = |PF(Zq(K))| |Bk

2 |

1/k .

(Bedlewo 2014) M and M∗ estimates July 16, 2014 15 / 36

Mean width: bound for vk(Zq(K))

Projections of PF(µ)

For any F ∈ Gn,k we have PF(Zq(µ)) = Zq(πF(µ)).

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Mean width: bound for vk(Zq(K))

Projections of PF(µ)

For any F ∈ Gn,k we have PF(Zq(µ)) = Zq(πF(µ)). Since πF(µ) is an isotropic log-concave measure on F we may use:

Theorem (Paouris)

If ν is an isotropic log-concave measure on Rk then vrad(Zq(ν)) c5 √q if q k and vrad(Zq(ν)) c5(q/k) √ k if q k

(Bedlewo 2014) M and M∗ estimates July 16, 2014 16 / 36

Mean width: bound for vk(Zq(K))

Projections of PF(µ)

For any F ∈ Gn,k we have PF(Zq(µ)) = Zq(πF(µ)). Since πF(µ) is an isotropic log-concave measure on F we may use:

Theorem (Paouris)

If ν is an isotropic log-concave measure on Rk then vrad(Zq(ν)) c5 √q if q k and vrad(Zq(ν)) c5(q/k) √ k if q k Applied to µ = µK, this gives vk(Zq(K)) c6 q k max(√q, √ k)LK.

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Mean width: E. Milman’s bound

Theorem (E. Milman)

For every centrally symmetric isotropic convex body K in Rn and for every 2 q n, M∗(Zq(K)) c√q(log q)2LK. In particular, M∗(K) c√n(log n)2LK.

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Mean width: E. Milman’s bound

Theorem (E. Milman)

For every centrally symmetric isotropic convex body K in Rn and for every 2 q n, M∗(Zq(K)) c√q(log q)2LK. In particular, M∗(K) c√n(log n)2LK. For the proof we write L−1

K

√nM∗(Zq(K)) L−1

K n

• k=1

1 √ k vk(zq(K)) c2

n

• k=1

max q k , q k

• ≃ q

q

• k=1

1 k + √q

n

• k=q

1 √q ≃ q log q + √q√n √n√q log q.

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Mean-norm: the dual problem

Let K be a centrally symmetric isotropic convex body K in Rn. In

• rder to estimate M(K) we may start from the estimate of V. Milman

and Pisier, using duality: we have √nM(K) c2

n

• k=1

1 √ k Radk(K ◦)vk(K ◦).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 18 / 36

Mean-norm: the dual problem

Let K be a centrally symmetric isotropic convex body K in Rn. In

• rder to estimate M(K) we may start from the estimate of V. Milman

and Pisier, using duality: we have √nM(K) c2

n

• k=1

1 √ k Radk(K ◦)vk(K ◦). Note that vk(K ◦) := sup{vrad(PF(K ◦)) : F ∈ Gn,k} ≃ 1 inf{vrad(K ∩ F) : F ∈ Gn,k} =: 1 w−

k (K).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 18 / 36

Mean-norm: the dual problem

Let K be a centrally symmetric isotropic convex body K in Rn. In

• rder to estimate M(K) we may start from the estimate of V. Milman

and Pisier, using duality: we have √nM(K) c2

n

• k=1

1 √ k Radk(K ◦)vk(K ◦). Note that vk(K ◦) := sup{vrad(PF(K ◦)) : F ∈ Gn,k} ≃ 1 inf{vrad(K ∩ F) : F ∈ Gn,k} =: 1 w−

k (K).

Because of the formula PF(Zq(µ)) = Zq(πF(µ)) we would prefer to work with the quantity v−

k (K) := inf{vrad(PF(K)) : F ∈ Gn,k}.

Note that w−

k (K) v− k (K).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 18 / 36

Mean norm: new general bound

Theorem (G.-E. Milman)

For every centrally symmetric convex body K in Rn and k 1 we have ek(K ◦, Bn

2 ) C n

k log

• e + n

k

• sup

1mmin(k,n)

1 2

k 3m v−

m(K)

.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 19 / 36

Mean norm: new general bound

Theorem (G.-E. Milman)

For every centrally symmetric convex body K in Rn and k 1 we have ek(K ◦, Bn

2 ) C n

k log

• e + n

k

• sup

1mmin(k,n)

1 2

k 3m v−

m(K)

. This leads to the next general bound:

Theorem (G.-E. Milman)

Let K be a centrally symmetric convex body in Rn with K ⊇ rBn

2 . Then,

√nM(K) C

n

• k=1

1 √ k min 1 r , n k log

• e + n

k

• 1

v−

k (K)

• .

(Bedlewo 2014) M and M∗ estimates July 16, 2014 19 / 36

Mean norm of Zq(µ)

We need a lower bound for vrad(PF(Zq(µ))) = vrad(Zq(πF(µ))), when F ∈ Gn,k.

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Mean norm of Zq(µ)

We need a lower bound for vrad(PF(Zq(µ))) = vrad(Zq(πF(µ))), when F ∈ Gn,k. The main tool is a theorem of Klartag and E. Milman: if ν is an isotropic log-concave measure on Rk then, for all 1 q √ k, vrad(Zq(ν)) c√q.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 20 / 36

Mean norm of Zq(µ)

We need a lower bound for vrad(PF(Zq(µ))) = vrad(Zq(πF(µ))), when F ∈ Gn,k. The main tool is a theorem of Klartag and E. Milman: if ν is an isotropic log-concave measure on Rk then, for all 1 q √ k, vrad(Zq(ν)) c√q. It follows that if µ is an isotropic log-concave measure on Rn then, for any q 1 and k = 1, . . . , n we have v−

k (Zq(µ)) c

• min(q,

√ k).

(Bedlewo 2014) M and M∗ estimates July 16, 2014 20 / 36

Mean norm of Zq(µ)

Theorem (G.-E. Milman)

For any isotropic log-concave measure µ on Rn and any q q0 := (n log(e + n))2/5, M(Zq(µ)) C √log q

4

√q .

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Mean norm of Zq(µ)

Theorem (G.-E. Milman)

For any isotropic log-concave measure µ on Rn and any q q0 := (n log(e + n))2/5, M(Zq(µ)) C √log q

4

√q . Since M(K) M(Zq0(K)) we have:

Theorem (G.-E. Milman)

For any centrally symmetric isotropic convex body K in Rn we have M(K) C LK log2/5(e + n) n1/10 .

(Bedlewo 2014) M and M∗ estimates July 16, 2014 21 / 36

One more problem

Paouris has proved that if µ is an isotropic log-concave measure on Rn, then Lµ ≃ C√n vrad(Zn(µ)) C√n vrad(Zq(µ)) for all q n. Klartag and E. Milman have proved that vrad(Z√n(µ)) c 4 √n. This gives a proof of the best known bound Lµ C 4 √n.

Question

To prove that M∗(Zq(µ)) c√q for q ≫ √n. By Urysohn’s inequality this is “less” than showing that vrad(Zq(µ)) c√q for q ≫ √n.

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Applications: I. Sub-Gaussian directions

A direction θ ∈ Sn−1 is a ψα-direction (where 1 α 2) for K with constant b > 0 if ·, θψα b·, θ2, where ·, θψα := inf

• t > 0 :
• K

exp |x, θ| t α dx 2

• .

It is known that ·, θψα ≃ sup

qα

·, θq q1/α .

(Bedlewo 2014) M and M∗ estimates July 16, 2014 23 / 36

Applications: I. Sub-Gaussian directions

From the Brunn-Minkowski inequality it follows that every θ ∈ Sn−1 is a ψ1-direction for K with an absolute constant C. Question: is it true that there exists an absolute constant C > 0 such that every K has at least one sub-Gaussian direction (ψ2-direction) with constant C? It is known that the answer is affirmative, with a constant O(√log n). This is due to Paouris, Valettas and G. (2011). The first result of this type was proved by Klartag (2006). In the isotropic case: is it true that most θ ∈ Sn−1 are sub-Gaussian directions for K with a constant at most logarithmic in n?

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Applications: I. Sub-Gaussian directions

Theorem (Brazitikos-Hioni)

Let K be an isotropic symmetric convex body in Rn. Then,

• Sn−1 ·, θψ2dσ(θ) C(log n)3LK,

where C > 0 is an absolute constant.

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Applications: I. Sub-Gaussian directions

Theorem (Brazitikos-Hioni)

Let K be an isotropic symmetric convex body in Rn. Then,

• Sn−1 ·, θψ2dσ(θ) C(log n)3LK,

where C > 0 is an absolute constant.

Theorem (Brazitikos-Hioni)

Let K be an isotropic symmetric convex body in Rn. Then, for any α > 1 we have ·, θψ2 C√α(log n)3/2 max √log nLK √α , L2

K

• for all θ in a subset Θ of Sn−1 with σ(Θ) 1 − n−α, where C > 0 is an

absolute constant.

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Applications: I. Sub-Gaussian directions

Proof of the first theorem: for any θ ∈ Sn−1 we have ·, θψ2 C1 max

1sm

hZ2s (K)(y) 2s/2 where m = ⌊log2 n⌋. It trivially follows that Eθ(·, θψ2) C1

m

• s=1

M∗(Z2s(K)) 2s/2 . We know that M∗(Z2s(K)) C2s2s/2 max

• s2s/2

√n , 1

• LK.

Therefore, Eθ(·, θψ2) C3

m

• s=1

s max

• s2s/2

√n , 1

• LK

C3m3LK C4(log n)3LK.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 26 / 36

Applications: II. Rogers-Shephard inequality

Let K be a convex body of volume 1 in Rn with 0 ∈ int(K). For every 1 k n − 1 and any F ∈ Gn,k we define g(K, k; F) :=

• |PF(K)| |K ∩ F ⊥|

1/k, where F ⊥ denotes the orthogonal subspace of F in Rn. A classical inequality of Rogers and Shephard states that if K is origin symmetric then 1 g(K, k; F) n k 1/k cn k , where c > 0 is an absolute constant. Both estimates are sharp.

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Applications: II. Rogers-Shephard inequality

Theorem (G.-Markessinis-Tsolomitis)

Let K be an isotropic convex body in Rn. For every 1 k n − 1 a random F ∈ Gn,k satisfies c1L−1

K

• n/k g(K, k; F) c2
• n/k(log n)2LK

with probability greater than 1 − e−k, where c1, c2 > 0 are absolute constants.

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Applications: II. Rogers-Shephard inequality

For the proof we show that, for any centered convex body K of volume 1 in Rn, For any 1 k n − 1 we have

• Gn,k

1 |PF(K)| |K ∩ F ⊥|dνn,k(F)

• c1

√ k √n k

Gn,k

1 |K ∩ F ⊥|

n n−k dνn,k(F)

n−k

n

, where c1 > 0 is an absolute constant. For any 1 k n − 1 we have

• Gn,k
• |PF(K)| |K ∩ F ⊥|

1/2 dνn,k(F)

• c2w(K)

√ k k/2 , where c2 > 0 is an absolute constant.

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Applications: II. Rogers-Shephard inequality

These two inequalities are general. However, for an isotropic convex body K in Rn we know that: For any 1 k n − 1 and any F ∈ Gn,k, |K ∩ F ⊥| c3 LK k , where c3 > 0 is an absolute constant. By E. Milman’s theorem, w(K) c4 √n(log n)2LK, where c4 > 0 is an absolute constant. This additional information completes the proof of the Theorem.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 30 / 36

Applications: III. Random polytopes

Let µ be an isotropic log-concave measure on Rn. For every N n we consider N independent random points x1, . . . , xN distributed according to µ and define the random polytope KN := conv{±x1, . . . , ±xN}. It was proved by Dafnis, G. and Tsolomitis that one can compare KN with the Lq-centroid body of µ for a suitable value of q; roughly speaking, KN is close to the body Zlog(N/n)(µ) with high probability.

(Bedlewo 2014) M and M∗ estimates July 16, 2014 31 / 36

Applications: III. Random polytopes

More precisely, given any isotropic log-concave measure µ on Rn and any cn N en, the random polytope KN satisfies, with high probability, the inclusion KN ⊇ c1Zlog(N/n)(µ). On the other hand, for every α > 1 and q 1, E

• σ({θ : hKN(θ) αhZq(µ)(θ)})
• Nα−q.

This estimate is sufficient for some sharp upper bounds: for all n N exp(n) one has E

• w(KN)
• c6 w(Zlog N(µ)).

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Applications: III. Random polytopes

For every 1 k n consider the normalized quermassintegrals of a convex body K: Qk(K) = Wn−k(K) ωn 1/k =

• 1

ωk

• Gn,k

|PF(K)| dνn,k(F) 1/k .

Theorem (Dafnis-G.-Tsolomitis)

If n2 N exp(cn) then for every 1 k n we have L−1

µ

• log N E
• Qk(KN)
• w(Zlog N(K)).

In the range n2 N exp(√n) one has an asymptotic formula: for every 1 k n, E

• Qk(KN)
• ≃
• log N.

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Applications: III. Random polytopes

Theorem (G.-Hioni-Tsolomitis)

Let x1, . . . , xN be independent random points distributed according to an isotropic log-concave measure µ on Rn, and consider the random polytope KN := conv{±x1, . . . , ±xN}. For all exp(√n) N exp(n) and s 1 we have L−1

µ

• log N Qk(KN) c2(s)
• log N (log log N)2,

for all 1 k < n, with probability greater than 1 − N−s.

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Applications: III. Random polytopes

For any convex body C in Rn and any 1 k n, the k-th mean outer radius of C is defined by ˜ Rk(C) =

• Gn,k R(PF(C)) dνn,k(F).

Theorem (Alonso-Guti´ errez, Dafnis, Hern´ andez-Cifre and Prochno)

If n N exp(√n) then a random KN satisfies, for all 1 k n, c max √ k,

• log(N/n)
• ˜

Rk(KN) C max √ k,

• log N
• .

Theorem (G.-Hioni-Tsolomitis)

If exp(√n) N exp(n) then a random KN satisfies, for all 1 k n, c max

• L−1

µ

• log N,

√ k,

• k/nR(Zlog N(µ))
• ˜

Rk(KN) C max{

• log N(log log N)2,
• k/n log N}.

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Applications: III. Random polytopes

Consider the average diameter of k-dimensional sections of a convex body C with 0 ∈ int(C), defined by ˜ Dk(C) =

• Gn,k R(C ∩ F) dνn,k(F).

Theorem (G.-Hioni-Tsolomitis)

Given 0 < a < b < 1, for any an k bn, a random KN satisfies with probability greater than 1 − exp(−ca √n): (i) If n2 N exp(√n) then ca max √log N log2 n , 1

• ˜

Dk(KN) cb

• log N,

(ii) If exp(√n) N exp(n) then ca √log N log2 n ˜ Dk(KN) cb

• log N(log log N)2,

where ca, cb > 0 depend only on a and b respectively.

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