On the intrinsic volumes of intersections of congruent balls

Bezdek, Károly

doi:10.1016/j.disopt.2019.03.002

Cited by 3 publications

(6 citation statements)

References 31 publications

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“…The statement that follows is a strengthening of as well as of [, Lemma 2.2], i.e., of [, (13)] and it can be derived from a volumetric inequality of Schramm in a rather straightforward way. For the sake of completeness, recall that

Q : = false{q_{1}, \dots, q_{N} false} \subset {double-struckE}^{d}

such that

false| {boldq}_{i} - {boldq}_{j} false| \leq λ holds for all 1 \leq i < j \leq N

, where

N \geq {2.359}^{d}

with d being sufficiently large.…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…and [8]). In this paper, we improve the later result for k = 0 in large dimensions moreover, extend Theorem 2 and Remark 3 to intrinsic volumes when K is a Euclidean ball in…”

Section: Introductionmentioning

confidence: 90%

“…If r ≤ cr(P), then V d (P r ) = V d (∅) = 0 and so, V d (P r ) ≤ V d (Q r ), i.e., (8) follows. Thus, for the rest of the proof we assume that cr(P ) < r, which together with Lemma…”

Section: Proof Of Part (Ii)mentioning

confidence: 99%

“…Next, as Euclidean balls are generating sets therefore implies the following statement. (See also [, (18); , Lemma 2.6].) Lemma If

d > 1

λ > 0

r > 0

, and

card (P) = N > 1

, then

V_{d} false(P^{r} false) \leq V_{d} false(B^{d} [o, r - false(N^{\frac{1}{d}} - 1 false) false(\frac{λ}{2} false)] false)

. Here we follow the convention that if

r - false(N^{\frac{1}{d}} - 1 false) false(\frac{λ}{2} false) < 0

, then

{boldB}^{d} false[boldo, r - (N^{\frac{1}{d}} - 1) (\frac{λ}{2}) false] = \emptyset

with

V_{d} false(\emptyset false) = 0

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We note that if

boldK

is a Euclidean ball in

R^{d}

, then Theorem improves [, Theorem 1.5] by replacing the condition

N \geq c^{d} d^{2.5 d}

with the weaker condition

N \geq 2^{d}

. On the other hand, [, Theorem 1.4] improves Theorem for

{double-struckM}_{boldK}^{d} = {double-struckE}^{d}

as follows: if

λ > 0

r > 0

d > 1

0 \leq k < d

N \geq {(1 + \sqrt{2})}^{d} = {(2.414 \dots)}^{d}

and

Q : = false{q_{1}, \dots, q_{N} false} \subset {double-struckE}^{d}

is a uniform contraction of

P : = false{p_{1}, \dots, p_{N} false} \subset {double-struckE}^{d}

with separating value λ in

E^{d}

, then

W_{k} false(P^{r} false) \leq W_{k} false(Q^{r} false)

(see also ). In this paper, we improve the later result for

k = 0

in large dimensions moreover, extend Theorem and Remark to intrinsic volumes when

boldK

is a Euclidean ball in

R^{d}

.…”

Section: Introductionmentioning

confidence: 99%

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On Uniform Contractions of Balls in Minkowski Spaces

Bezdek

2020

Mathematika

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Let N balls of the same radius be given in a d-dimensional real normed vector space, i.e., in a Minkowski d-space. Then apply a uniform contraction to the centers of the N balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (respectively, decrease) the volume of the union (respectively, intersection) of N balls in Minkowski d-space, provided that N 2 d (respectively, N 3 d and the unit ball of the Minkowski d-space is a generating set). Some improvements are presented in Euclidean spaces. §1. Introduction. The Kneser-Poulsen conjecture [17,26] (respectively, Gromov-Klee-Wagon conjecture [12,15,16]) states that if the centers of a family of N unit balls in Euclidean d-space is contracted, then the volume of the union (respectively, intersection) does not increase (respectively, decrease). These conjectures have been proved by Bezdek and Connelly [5] for d = 2 (in fact, for not necessarily congruent circular disks as well) and they are open for all d 3. For a number of partial results in dimensions d 3, we refer the interested reader to the corresponding chapter in [3]. Very recently Bezdek and Naszódi [8] investigated the Kneser-Poulsen conjecture as well as the Gromov-Klee-Wagon conjecture for special contractions in particular, for uniform contractions. Here, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main result of [8] states that a uniform contraction of the centers does not increase (respectively, decrease) the volume of the union (respectively, intersection) of N unit balls in Euclidean d-space (d 3), provided that N c d d 2.5d , where c > 0 is a universal constant (respectively, N (1 + √ 2) d ). In this paper we improve these results and extend them to Minkowski spaces.Let K ⊂ R d be an o-symmetric convex body, i.e., a compact convex set with non-empty interior symmetric about the origin o in R d . Let · K denote the norm generated by K, which is defined by x K := min{λ 0 | λx ∈ K} for x ∈ R d . Furthermore, let us denote R d with the norm · K by M d K and call it the Minkowski space of dimension d generated by K. We write B K [x, r] := x + rK for x ∈ R d and r > 0 and call any such set a (closed) ball of radius r, where + refers to vector addition extended to subsets of R d in the usual way. The following definitions introduce the core notions and notations for our paper. Definition 1. For X ⊆ R d and r > 0 let X K r := {B K [x, r] | x ∈ X } respectively, X r K := {B K [x, r] | x ∈ X }

show abstract

Q : = false{q_{1}, \dots, q_{N} false} \subset {double-struckE}^{d}

such that

false| {boldq}_{i} - {boldq}_{j} false| \leq λ holds for all 1 \leq i < j \leq N

, where

N \geq {2.359}^{d}

with d being sufficiently large.…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…and [8]). In this paper, we improve the later result for k = 0 in large dimensions moreover, extend Theorem 2 and Remark 3 to intrinsic volumes when K is a Euclidean ball in…”

Section: Introductionmentioning

confidence: 90%

“…If r ≤ cr(P), then V d (P r ) = V d (∅) = 0 and so, V d (P r ) ≤ V d (Q r ), i.e., (8) follows. Thus, for the rest of the proof we assume that cr(P ) < r, which together with Lemma…”

Section: Proof Of Part (Ii)mentioning

confidence: 99%

“…Next, as Euclidean balls are generating sets therefore implies the following statement. (See also [, (18); , Lemma 2.6].) Lemma If

d > 1

λ > 0

r > 0

, and

card (P) = N > 1

, then

V_{d} false(P^{r} false) \leq V_{d} false(B^{d} [o, r - false(N^{\frac{1}{d}} - 1 false) false(\frac{λ}{2} false)] false)

. Here we follow the convention that if

r - false(N^{\frac{1}{d}} - 1 false) false(\frac{λ}{2} false) < 0

, then

{boldB}^{d} false[boldo, r - (N^{\frac{1}{d}} - 1) (\frac{λ}{2}) false] = \emptyset

with

V_{d} false(\emptyset false) = 0

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We note that if

boldK

is a Euclidean ball in

R^{d}

, then Theorem improves [, Theorem 1.5] by replacing the condition

N \geq c^{d} d^{2.5 d}

with the weaker condition

N \geq 2^{d}

. On the other hand, [, Theorem 1.4] improves Theorem for

{double-struckM}_{boldK}^{d} = {double-struckE}^{d}

as follows: if

λ > 0

r > 0

d > 1

0 \leq k < d

N \geq {(1 + \sqrt{2})}^{d} = {(2.414 \dots)}^{d}

and

Q : = false{q_{1}, \dots, q_{N} false} \subset {double-struckE}^{d}

is a uniform contraction of

P : = false{p_{1}, \dots, p_{N} false} \subset {double-struckE}^{d}

with separating value λ in

E^{d}

, then

W_{k} false(P^{r} false) \leq W_{k} false(Q^{r} false)

(see also ). In this paper, we improve the later result for

k = 0

in large dimensions moreover, extend Theorem and Remark to intrinsic volumes when

boldK

is a Euclidean ball in

R^{d}

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Uniform Contractions of Balls in Minkowski Spaces

Bezdek

2020

Mathematika

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show abstract

Volumetric bounds for intersections of congruent balls in Euclidean spaces

Bezdek

2021

Aequat. Math.

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On Blaschke–Santaló-Type Inequalities for r-Ball Bodies

Bezdek

2023

MathPann

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Let 𝔼𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

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On the intrinsic volumes of intersections of congruent balls

Cited by 3 publications

References 31 publications

On Uniform Contractions of Balls in Minkowski Spaces

On Uniform Contractions of Balls in Minkowski Spaces

Volumetric bounds for intersections of congruent balls in Euclidean spaces

On Blaschke–Santaló-Type Inequalities for r-Ball Bodies

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