Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem - - PowerPoint PPT Presentation

Background Sublinear Upper Bounds Lower Bounds Final remarks

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem

Stephen R. Chestnut Robert Hildebrand* Rico Zenklusen

Institute for Operations Research, ETH Z¨ urich & IBM T.J. Watson Research Center

January 7, 2016

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Presentation Outline

1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Helly (1914) Let C be a collection of compact convex sets in Rd. If

(d+1)

• i=1

Ci = ∅ ⇒

• C∈C

C = ∅.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Helly (1914) Doignon (1973) Let C be a collection of compact convex sets in Rn. If

2n

• i=1

Ci∩Zn = ∅ ⇒

• C∈C

C∩Zn = ∅.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Helly (1914) Doignon (1973) Averkov-Weismantel (2012) Let C be a collection of compact convex sets in Rd+n. If

(d+1)2n

• i=1

Ci ∩ (Rd × Zn) = ∅ ⇒

• C∈C

C ∩ (Rd × Zn) = ∅.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

• I. Aliev, R. Bassett, J. A. De Loera, and Q. Louveaux, A quantitative

Doignon-Bell-Scarf theorem, (2014)

• G. Averkov, On maximal S-free sets and the Helly number for S-convex sets,

(2013)

• G. Averkov and R. Weismantel, Transversal numbers over subsets of linear

spaces, (2012)

• G. Kalai and R. Meshulam, A topological colorful Helly theorem, (2005)
• G. Kalai and R. Meshulam, Leray numbers of projections and a topological

Helly-type theorem,(2008)

• C. Knauer, H. R. Tiwary, and D. Werner, On the computational complexity of

ham-sandwich cuts, Helly sets, and related problems, (2011)

• L. Montejano and D. Oliveros, Colourful transversal theorems, (2008)
• Montejano and D. Oliveros, Tolerance in Helly-type theorems, (2011)
• L. Montejano, A New Topological Helly Theorem and Some Transversal Results,

(2014)

• P. Sober´
• n, Quantitative (p, q) theorems in combinatorial geometry, (2015)
• P. Sober´
• n, Helly-type theorems for the diameter, (2015)
• K. Swanepoel, Helly-type theorems for homothets of planar convex

curves,(2003)

• R. Wenger, Helly-type theorems and geometric transversals, (2004)

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Discrete Quantitative Theorem

Definition

c(n, k) is the least integer such that for any A ∈ Rm×n, and b ∈ Rm, if {x ∈ Rn : Ax ≤ b} has exactly k integer solutions, then there exits a subset S of the rows of A with |S| ≤ c(n, k) such that {x ∈ Rn : ASx ≤ bS} has exactly k integer solutions.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Discrete Quantitative Theorem

Definition

c(n, k) is the least integer such that for any A ∈ Rm×n, and b ∈ Rm, if {x ∈ Rn : Ax ≤ b} has exactly k integer solutions, then there exits a subset S of the rows of A with |S| ≤ c(n, k) such that {x ∈ Rn : ASx ≤ bS} has exactly k integer solutions.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Discrete Quantitative Theorem

Definition

c(n, k) is the least integer such that for any A ∈ Rm×n, and b ∈ Rm, if {x ∈ Rn : Ax ≤ b} has exactly k integer solutions, then there exits a subset S of the rows of A with |S| ≤ c(n, k) such that {x ∈ Rn : ASx ≤ bS} has exactly k integer solutions.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Discrete Quantitative Theorem

Definition

c(n, k) is the least integer such that for any A ∈ Rm×n, and b ∈ Rm, if {x ∈ Rn : Ax ≤ b} has exactly k integer solutions, then there exits a subset S of the rows of A with |S| ≤ c(n, k) such that {x ∈ Rn : ASx ≤ bS} has exactly k integer solutions.

Corollary (ABDL ’14)

Let C be a finite collection of compact convex sets in Rn. If

• c(n,k)
• i=1

Ci ∩ Zn

• ≥ k + 1

⇒

• C∈C

C ∩ Zn

• ≥ k + 1.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bounds on c(n, k)

Upper bounds

1 Bell (1977):

c(n, k) ≤ (k + 2)n

2 ABDL (2014):

c(n, k) ≤ ⌈2(k + 1)/3⌉2n − 2⌈2(k + 1)/3⌉ + 2

3 This talk: c(n, k) = o(k)2n

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bounds on c(n, k)

Upper bounds

1 Bell (1977):

c(n, k) ≤ (k + 2)n

2 ABDL (2014):

c(n, k) 2

3k2n 3 This talk: c(n, k) = o(k)2n

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bounds on c(n, k)

Upper bounds

1 Bell (1977):

c(n, k) ≤ (k + 2)n

2 ABDL (2014):

c(n, k) 2

3k2n 3 This talk: c(n, k) = o(k)2n 4 Bell (1977):

c(2, k) = O(k

1 3 ) Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bounds on c(n, k)

Upper bounds

1 Bell (1977):

c(n, k) ≤ (k + 2)n

2 ABDL (2014):

c(n, k) 2

3k2n 3 This talk: c(n, k) = o(k)2n 4 Bell (1977):

c(2, k) = O(k

1 3 )

Lower bounds

1 ??? c(2, k) = Ω(k

1 3 )

2 This talk: c(n, k) = Ωn(k

n−1 n+1 ) Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Presentation Outline

1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bell’s expanding polyhedron lemma

⇒ Lemma: There exists a polytope with c(n, k) facets, exactly one integer point on each facet found on its relative interior, and containing exactly k integer points not on the boundary.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Bell’s expanding polyhedron lemma

⇒ Lemma: There exists a polytope with c(n, k) facets, exactly one integer point on each facet found on its relative interior, and containing exactly k integer points not on the boundary.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k,

Lemma

c(n, k) < ℓ(n, k + 1)

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k,

Lemma

c(n, k) < ℓ(n, k + 1)

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k, Theorem: [ABDL ’14] ℓ(n, k) ≤ ⌈2k/3⌉2n − 2⌈2k/3⌉ + 2

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k, Theorem: [ABDL ’14] ℓ(n, k) ≤ ⌈2k/3⌉2n − 2⌈2k/3⌉ + 2 Lemma: ℓ(n, 1) = 2n + 1.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k, Theorem: [ABDL ’14] ℓ(n, k) ≤ ⌈2k/3⌉2n − 2⌈2k/3⌉ + 2 Lemma: ℓ(n, 1) = 2n + 1. proof: [≤] Consider the 2n parities {even/odd}n, i.e., x (mod 2). By pigeonhole principle, ∃ parity with 2 points x, y. Since x ≡ y (mod 2), the midpoint 1

2(x + y) ∈ Zn is integral.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k, Theorem: [ABDL ’14] ℓ(n, k) ≤ ⌈2k/3⌉2n − 2⌈2k/3⌉ + 2 Lemma: ℓ(n, 1) = 2n + 1. Simple Lemma: ℓ(n, k) ≤ k2n + 1

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Convex hull of lattice points

Definition

ℓ(n, k) is the least integer such that for any S ⊆ Zn with |S| ≥ ℓ(n, k) and S in convex position, |(conv(S) ∩ Zn) \ S| ≥ k, Theorem: [ABDL ’14] ℓ(n, k) ≤ ⌈2k/3⌉2n − 2⌈2k/3⌉ + 2 Lemma: ℓ(n, 1) = 2n + 1. Simple Lemma: ℓ(n, k) ≤ k2n + 1 proof: [≤] Consider the 2n parities {even/odd}n, i.e., x (mod 2). By pigeonhole principle, ∃ parity with k + 1 points. There are at least k distinct midpoints within this parity, which are all integral.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn.

Lemma

Let n ≥ 1, k ≥ 0, and let µ−1

c (n, k) := min{s : µc(n, s) ≥ k}.

Therefore, c(n, k) ≤ ℓ(n, k + 1) ≤ µ−1

c (n, k + 1) · 2n.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn.

Lemma

Let n ≥ 1, k ≥ 0, and let µ−1

c (n, k) := min{s : µc(n, s) ≥ k}.

Therefore, c(n, k) ≤ ℓ(n, k + 1) ≤ µ−1

c (n, k + 1) · 2n.

Example: µc(2, s) ≥ 1

4s2

⇒ c(n, k) ≤ 8 √ k + 1.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn. Lemma [Erd¨

• s, Fishburn, F¨

uredi (’91)]: For all s ≥ 3 and n = 2 s 2

• −
• s(s + 1)(1 − e−1/2)

4

• ≤ µc(2, s) ≤

s 2

• −

s2 − 2s + 12 20

• .

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn. Lemma [Erd¨

• s, Fishburn, F¨

uredi (’91)]: For all s ≥ 3 and n = 2 s 2

• −
• s(s + 1)(1 − e−1/2)

4

• ≤ µc(2, s) ≤

s 2

• −

s2 − 2s + 12 20

• .

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn. Lemma [Erd¨

• s, Fishburn, F¨

uredi (’91)]: For all s ≥ 3 and n = 2 s 2

• −
• s(s + 1)(1 − e−1/2)

4

• ≤ µc(2, s) ≤

s 2

• −

s2 − 2s + 12 20

• .

No known extension to higher dimensions....

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn. Theorem [Sanders (’10)]: ∃ constant C s.t. for any abelian group G, finite subset A ⊆ G with no three-term arithmetic progressions |A + A| ≥ C|A| log1/3 |A| log log |A|.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Midpoints

Definition

µc(n, s) is the least number of distinct midpoints possible given s points in convex position in Rn. Theorem [Sanders (’10)]: ∃ constant C s.t. for any abelian group G, finite subset A ⊆ G with no three-term arithmetic progressions |A + A| ≥ C|A| log1/3 |A| log log |A|.

Corollary (Sublinear in k)

For all n, k ≥ 1, c(n, k) ≤ ℓ(n, k + 1) ≤ O

• k log log k

log1/3 k

• · 2n.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Minimum Area of v-gons and Pick’s Theorem

A lattice polygon with v vertices, b boundary lattice points, and i interior lattice points satisfies A = i + v + b 2 − 1 By [Rabinowitz ’93], the minimum area A ≥

v3 8π2 . Hence

v3 8π2 − v 2 ≤ i + b 2 − 1 < k. Using a lower bound, this implies, for instance that c(2, k) ≤ ℓ(2, k + 1) ≤ ⌈4.43(k + 4)

1 3 ⌉.

See also [B´ ar´ any and Tokushige ’04] for more precise asymptotic constants.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Presentation Outline

1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points. From left to right, these examples show that α(2, 2) ≥ 6, α(2, 4) ≥ 8, and α(2, 5) ≥ 7.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k). proof:

1 Let S be maximal for α(n, k).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k). proof:

1 Let S be maximal for α(n, k). 2 Let P have a facet through each

point in S containing conv(S).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k). proof:

1 Let S be maximal for α(n, k). 2 Let P have a facet through each

point in S containing conv(S).

3 If int(P) ∩ Zn > k, then S was not

a maximal example for α(n, k).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k). proof:

1 Let S be maximal for α(n, k). 2 Let P have a facet through each

point in S containing conv(S).

3 If int(P) ∩ Zn > k, then S was not

a maximal example for α(n, k).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Definition

α(n, k):= max {|S| : S ⊆ Zn in convex position, | ic(S) \ S| = k} , i.e., the maximum number of vertices of a lattice polytope containing exactly k other lattice points.

Lemma

For all n ≥ 1 and k ≥ 0, c(n, k) ≥ α(n, k). proof:

1 Let S be maximal for α(n, k). 2 Let P have a facet through each

point in S containing conv(S).

3 If int(P) ∩ Zn > k, then S was not

a maximal example for α(n, k).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

c(n, 1) = c(n, 2) = 2(2n − 1)

By [ABDL ’14],

1 c(n, 1) ≤ 2(2n − 1), 2 c(n, 2) ≤ 2(2n − 1), and 3 c(n, 1) ≥ 2(2n − 1).

Example 1: α(n, 1) ≥ 2(2n − 1). Consider ({−1, 0}n ∪ {0, 1}n) \ {0}n. Example 2: α(n, 2) ≥ 2(2n − 1). Consider ({−1, 0}n ∪{0, 1}n ∪{ 2})\{ 0, 1}

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Induction tool

Lemma

For all k, n ≥ 1, c(n, k) ≥ c(n, k − 1) − 1. Let k − 1 = 4.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Induction tool

Lemma

For all k, n ≥ 1, c(n, k) ≥ c(n, k − 1) − 1. Let k − 1 = 4.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Induction tool

Lemma

For all k, n ≥ 1, c(n, k) ≥ c(n, k − 1) − 1. Let k − 1 = 4.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Lower Bound with Spheres

Theorem

For each n ≥ 2, there exists a constant C, depending only on n, such that c(n, k) ≥ C k

n−1 n+1 .

proof:

1 Let Pr = conv(B(a, r) ∩ Zn). 2 # vertices ≈ r

n(n−1) n+1

3 # integer points total ≈ rn 4 ⇒ α(n, kr) ≥ Ckr

n−1 n+1

5 |k − kr| is “small” 6 c(n, k) ≥ c(n, k − 1) − 1 ⇒ c(n, k) ≥ C ′k

n−1 n+1 . Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Lower Bound with Spheres

Theorem

For each n ≥ 2, there exists a constant C, depending only on n, such that c(n, k) ≥ C k

n−1 n+1 .

proof:

1 Let Pr = conv(B(a, r) ∩ Zn). Choose a in (0, 1)n special 2 # vertices ≈ r

n(n−1) n+1

3 # integer points total ≈ rn 4 ⇒ α(n, kr) ≥ Ckr

n−1 n+1

5 |k − kr| is “small” 6 c(n, k) ≥ c(n, k − 1) − 1 ⇒ c(n, k) ≥ C ′k

n−1 n+1 . Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Presentation Outline

1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

c(n, k) is not monotonically increasing!

c(2, 4) = 8, but c(2, 5) = 7

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Upper bound for fewest number of midpoints µc

Corollary

For each n ∈ Z≥2, there exists a constant C such that the fewest number of distinct midpoints µc(n, s) of any s points in convex position in Rn is bounded as µc(n, s) ≤ C s

n+1 n−1 . Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Open Questions and other authors

Open questions

1 Asymptotic behavior of c(n, k), α(n, k), ℓ(n, k), and µc(n, s) 2 Relationship to h(Zn \ S) and f (Zn \ S)

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

Open Questions and other authors

Open questions

1 Asymptotic behavior of c(n, k), α(n, k), ℓ(n, k), and µc(n, s) 2 Relationship to h(Zn \ S) and f (Zn \ S)

Other authors

1 Ruben La Haye Ph.D. Thesis

(Student at UC Davis under Jes´ us De Loera) Computation of numbers c(2, k) for k ≤ 40, and more?

2 Friends at Magdeburg, T.U. Munich, and F.U. Berlin

Other work on c(n, k) in general dimensions.

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks

References

⋄ Stephen R. Chestnut, Robert Hildebrand, Rico Zenklusen. Sublinear Bounds for a Quantitative Doignon-Bell-Scarf Theorem. arxiv:1512.07126 [math.CO] (2015). ⋄ Iskander Aliev, Robert Bassett, Jes´ us A. De Loera, and Quentin Louveaux. A quantitative Doignon-Bell-Scarf theorem. arXiv:1405.2480 [math.CO] (2014), to appear in Combinatorica. ⋄ Nina Amenta, Jes´ us A. De Loera, and Pablo Sober´

• n.

Helly’s Theorem: New Variations and Applications arxiv:1508.0760 [math.mg] (2015).

Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen