Coverings in Euclidean space and on the Sphere
Márton Naszódi
École polytechnique fédérale
de Lausanne (Lausanne)
Eötvös University
(Budapest)
Covering Numbers
K , L – bounded Borel sets in Rn .
Definition
Translative covering number of K by L: The minimum number of
translates of L that cover K .
Denoted by N(K , L).
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Covering in Rn
Theorem (Rogers, ’57)
L ⊂ Rn a convex body. Then the covering density of L is at most
n ln n + n ln ln n + 5n.
Corollary (Rogers–Zong, ’97)
K , L ⊂ Rn convex bodies. Then
vol(K − L)
vol(K )
≤ N(K , L) ≤
(n ln n + n ln ln n + 5n).
vol(L)
vol L
Theorem (G. Fejes Tóth, ’09)
density ≤ n ln n + n ln ln n + n + o(n).
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Covering the Sphere by Caps
Theorem (Böröczky Jr. – Wintsche, ’03)
Let 0 < ϕ < π2 . Then there is a covering of Sn by spherical caps of radius
ϕ with density at most n ln n + n ln ln n + 5n.
Theorem (Dumer, ’07)
For ϕ < π3 : density ≤
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n ln n
2 .
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Special Case: The Illumination Problem
Gohberg–Markus–Levi–Boltyanskii–Hadwiger Illumination Conjecture
Fix n. Then the maximum of N(K , int K ) over all convex bodies K in Rn
is 2n , and only attained by parallelotopes.
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Special Case: The Illumination Problem
Gohberg–Markus–Levi–Boltyanskii–Hadwiger Illumination Conjecture
Fix n. Then the maximum of N(K , int K ) over all convex bodies K in Rn
is 2n , and only attained by parallelotopes.
Known:
Rogers
(
i(K ) := N(K , int K ) ≤
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2n (n ln n + n ln ln n + 5n)
if K = −K ,
2n
n (n ln n + n ln ln n + 5n) otherwise.
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Fractional Covering Numbers
K , L – bounded Borel sets in Rn .
Recall:
Definition
Translative covering number of K by L: the minimum number of
translates of L that cover K .
Denoted by N(K , L).
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Fractional Covering Numbers
K , L – bounded Borel sets in Rn .
Recall:
Definition
Translative covering number of K by L: the minimum number of
translates of L that cover K .
Denoted by N(K , L).
Definition (MN, ’09; Artstein-Avidan – Raz, ’11)
Fractional covering of K by translates of L: a Borel measure µ on Rn with
µ(x − L) ≥ 1 ∀x ∈ K .
Fractional covering number:
N ∗ (K , L) = inf {µ(Rn ) : µ is a fract. covering of K by translates of L}
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(Fractional) Covering Numbers
MN, ’09
(
i ∗ (K ) := N ∗ (K , int K ) ≤
2n
2n
n
if K = −K ,
otherwise.
Artstein-Avidan – Slomka (and Schneider), ’13
In the symmetric case, equality is attained by parallelotopes only.
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(Fractional) Covering Numbers
MN, ’09
(
i ∗ (K ) := N ∗ (K , int K ) ≤
2n
2n
n
if K = −K ,
otherwise.
Artstein-Avidan – Slomka (and Schneider), ’13
In the symmetric case, equality is attained by parallelotopes only.
Connection between fractional and integral covering:
Artstein-Avidan – Slomka, ’13
K , L ⊆ Rn bounded, Borel sets. Λ⊂ K a finite δ-net of K w.r.t. L (ie.
K ⊆ δL + Λ). Let L−δL := {x ∈ L : x + δL ⊂ L}.
Then
N ∗ (K , L) ≤ N(K , L) ≤ ln 4|Λ|) N ∗ (K , L−δL ) +
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q
ln 4|Λ| N ∗ (K , L−δL ).
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Methods
Method 1. Rogers ’57, Erdős–Rogers ’61, Böröczky Jr. – Wintsche ’03,
Dumer ’07, G. Fejes Tóth ’09:
Cover almost all of space randomly and then, cover the gaps.
Method 2a. Schramm ’88, Füredi–Kang ’08, Artstein-Avidan – Slomka
’13: Construct a δ-net, Λ. Cover Λ randomly by a slightly smaller copy of
L, say by L−δL := {x ∈ L : x + δL ⊂ L}. Finally, replace translates of
L−δL by L. If δ was small enough, then the density does not increase much
at the last step.
Method 2b. MN, ’14: Construct a δ-net. Use a combinatorial lemma of
Lovász to cover Λ by a slightly smaller copy of L, . . . .
2a/2b differences: non-probabilistic, short proofs.
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Main result
Theorem
K , L ⊆ Rn bounded, Borel sets. Λ⊂ Rn a finite δ-net of K w.r.t. L (ie.
K ⊆ δL + Λ). Then
N(K , L) ≤ 1 + ln
max
x ∈K −L−δL
card (x + L−δL ) ∩ Λ
· N ∗ (K − δL, L−δL ).
If Λ ⊂ K then we have
N(K , L) ≤ 1 + ln
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max
x ∈K −L−δL
card (x + L−δL ) ∩ Λ
· N ∗ (K , L−δL ).
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Proof of the main result
Lemma (Lovász, ’75)
Λ finite set, H ⊆ 2Λ . Then the greedy algorithm yields
τ (Λ, H) ≤ (1 + ln(max card H))τ ∗ (Λ, H),
H∈H
where τ (Λ, H) is the covering number of Λ by H, and τ ∗ is the fractional
covering number.
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Proof of the main result
Lemma (Lovász, ’75)
Λ finite set, H ⊆ 2Λ . Then the greedy algorithm yields
τ (Λ, H) ≤ (1 + ln(max card H))τ ∗ (Λ, H),
H∈H
where τ (Λ, H) is the covering number of Λ by H, and τ ∗ is the fractional
covering number.
Observation
Let Y be a set, F ⊆ 2Y , X ⊆ Y .
Let Λ ⊆ U ⊆ Y , with Λ finite.
Assume that for some F 0 ⊆ 2Y we have τ (X , F) ≤ τ (Λ, F 0 ). Then
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Proof of the main result
Lemma (Lovász, ’75)
Λ finite set, H ⊆ 2Λ . Then the greedy algorithm yields
τ (Λ, H) ≤ (1 + ln(max card H))τ ∗ (Λ, H),
H∈H
where τ (Λ, H) is the covering number of Λ by H, and τ ∗ is the fractional
covering number.
Observation
Let Y be a set, F ⊆ 2Y , X ⊆ Y .
Let Λ ⊆ U ⊆ Y , with Λ finite.
Assume that for some F 0 ⊆ 2Y we have τ (X , F) ≤ τ (Λ, F 0 ). Then
τ (X , F) ≤ τ (Λ, F 0 ) ≤ (1 + ln( max
card{Λ ∩ F 0 })) · τ ∗ (U, F 0 ).
0
0
F ∈F
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Proof of the main result
Observation
Let Y be a set, F ⊆ 2Y , X ⊆ Y .
Let Λ ⊆ U ⊆ Y , with Λ finite.
Assume that for some F 0 ⊆ 2Y we have τ (X , F) ≤ τ (Λ, F 0 ). Then
τ (X , F) ≤ τ (Λ, F 0 ) ≤ (1 + ln( max
card{Λ ∩ F 0 })) · τ ∗ (U, F 0 ).
0
0
F ∈F
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Proof of the main result
Observation
Let Y be a set, F ⊆ 2Y , X ⊆ Y .
Let Λ ⊆ U ⊆ Y , with Λ finite.
Assume that for some F 0 ⊆ 2Y we have τ (X , F) ≤ τ (Λ, F 0 ). Then
τ (X , F) ≤ τ (Λ, F 0 ) ≤ (1 + ln( max
card{Λ ∩ F 0 })) · τ ∗ (U, F 0 ).
0
0
F ∈F
Now, substitute!
Y = Rn , X = K , F = {L + x : x ∈ Rn },
Λ: a δ-net of K w.r.t. L;
F 0 = {L−δL + x : x ∈ Rn }.
U = K − δL.
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Applications
Theorem
K ⊆ Rn a bounded Borel set. Then there is a covering of Rn by translated
copies of K of density at most


vol K−δ/2

vol(K ) 
 .
1 + ln
inf 
vol(K−δ )
vol B o, δ
δ>0
2
Corollary: Rogers’ bound for covering Rn by translates of a convex body.
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Applications
Theorem
K ⊆ Sn a Borel set. Then there is a covering of Sn by rotated copies of K
of density at most


σ(K ) 
inf 
1 + ln
δ>0 σ(K−δ )
σ K−δ/2
Ω
δ
2

 .
Corollary
K ⊆ Sn spherically convex, of spherical circumradius ρ. Then there is a
covering of Sn by rotated copies of K of density at most
inf
κ>0 : K−(κρ) 6=∅
1
σ(K )
2n + n ln
σ(K ) − Ω(ρ) (1 − (1 − κ)n )
κρ
.
Corollary: Böröczky Jr. – Wintsche bound for caps.
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