On the complements of union of open balls of xed radius in the Euclidean space Marco Longinetti

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On the complements of union of open balls of ﬁxed radius in the

Euclidean space

Marco Longinetti∗

Paolo Manselli†

Adriana Venturi‡

Abstract

Let an R-body be the complement of the union of open balls of radius Rin Ed. The R-hulloid

of a closed not empty set A, the minimal R-body containing A, is investigated; if Ais the set of

the vertices of a simplex, the R-hulloid of Ais completely described (if d= 2) and if d > 2 special

examples are studied. The class of R-bodies is compact in the Hausdorﬀ metric if d= 2, but not

compact if d > 2.

1 Introduction

Given a closed set E⊂Ed(d≥2), the convex hull of Eis the intersection of all closed half spaces

containing E; the convex hull can be considered as a regularization of E. Given R > 0, a diﬀerent

hull of Ecould be the intersection of all closed sets, containing E, complement of open balls of radius

Rnot intersecting E. Let us call this set the R-hulloid of E, denoted as coR(E); the R-bodies are

the sets coinciding with their R-hulloids. R-bodies are called 2R-convex sets in [10].

The R-hulloid coR(E) has been introduced by Perkal [10] as a regularization of E, hinting that

coR(E) is a mild regularization of a closed set. Mani-levitska [8] hinted that the R-bodies cannot be

too irregular.

In our work it is shown that this may not be true: in Theorem 5.8 an example of a connected

set is constructed with disconnected R-hulloid. A deeper study gave us the possibility to add new

properties to the R-bodies: a representation of coR(E) is given in Theorem 3.10 and new properties

of ∂coR(E) are proved in Theorem 3.12, Theorem 3.13 and Corollary 3.14. Moreover contrasting

results on regularity are found: every closed set contained in an hyperplane or in a sphere of radius

r≥Ris an R-body (Theorems 3.19 and 3.20). As a consequence a problem of Borsuk, quoted by

Perkal [10], has a negative answer (Remark 3.21). In §4 it is shown that the R-body regularity

heavily depends on the dimension. A deﬁnition (Deﬁnition 4.3) similar to the classic convexity is

given for the class of planar R-bodies, namely (Theorem 4.6):

Ais an R-body iﬀ coR({a1, a2, a3})⊂A∀a1, a2, a3∈A.

∗Marco.Longinetti@uniﬁ.it, Dipartimento DIMAI, Universit`a degli Studi di Firenze, V.le Morgagni 67/a, 50138

Firenze - Italy

†Paolo.Manselli@uniﬁ.it, Firenze, Italy

‡Adriana.Venturi@uniﬁ.it, Firenze, Italy

arXiv:2210.04276v1 [math.AP] 9 Oct 2022

As consequence, if d= 2: a sequence of compact R-bodies converges in the Hausdorﬀ metric to an

R-body (Corollary 4.8). If d > 2, in Theorem 3.16 it is proved that a sequence of compact R-bodies

converges to an (R−)-body, for every 0 <  < R; however, the body limit may not to be an R-body

as an example in §5 shows. If Eis connected, properties of connectivity of coR(E) are investigated

in §4.3.

In [7, Deﬁnition 2.1] V. Golubyatnikov and V. Rovenski introduced the class K1/R

2. In Theorem

6.1 it is proved that the class of R-bodies is strongly contained in K1/R

2. If d= 2 , under additional

assumptions, it is also proved that the two classes coincide.

2 Deﬁnitions and Preliminaries

Let Ed, d ≥2,be the linear Euclidean Space with unit sphere Sd−1;A⊂Edwill be called a body

if Ais non empty and closed. The minimal aﬃne space containing Awill be Lin(A). The convex

hull of Awill be co(A); for notations and results of convex bodies, let us refer to [13].

Deﬁnition 2.1. Let Abe a not empty set.

A:= {x∈Ed:dist(A, x)< ;};A0

:= {x∈Ed:dist(A, x)≥};A−:= A∪∂A;Ac:= Ed\A;

Int(A) = A−\∂A.

B(x, r)will be the open ball of center x∈Edand radius r > 0; a sphere of radius ris ∂B(x, r).

Proposition 2.2. Let us recall the following facts for reference.

•1Ais open; A= (A−)⊂(A)−.

•2A={x∈Ed:∃a∈A, for which x∈B(a, )}={x∈Ed:B(x, )∩A6=∅,}=

∪a∈AB(a, ) = A+B(0, ).

•3A0

={x∈Ed:∀a∈A,x /∈B(a, )}=

={x∈Ed:B(x, )∩A=∅} =∩a∈A{B(a, )c}.

•4Let Ai, i = 1,2be non empty sets. Then

A1⊂A2⇒(A1)⊂(A2).

•5If Eis non empty, then E⊂(E0

R)0

R⊂ER, see [1, lemma 4.3].

Deﬁnition 2.3. ([3]) If A⊂Ed, a ∈A, then reach(A, a)is the supremum of all numbers ρsuch

that for every x∈B(a, ρ)there exists a unique point b∈Asatisfying |b−x|= dist (x, A). Also:

reach(A) := inf{reach(A, a) : a∈A}.

Let b1, b2∈Ed,|b1−b2|<2Rand let h(b1, b2) be the intersection of all closed balls of radius R

containing b1, b2.

Proposition 2.4. ([1, Theorem 3.8], [11]) The body Ahas reach ≥Rif and only if A∩h(b1, b2)is

connected for every b1, b2∈A, 0<|b1−b2|<2R.

Remark 2.5. The R-hull of a set Ewas introduced in [1, Deﬁnition 4.1] as the minimal set ˆ

Eof

reach ≥Rcontaining E. Therefore if reach(A)≥R, then Acoincides with its R-hull. The R-hull

of a set E may not exist, see [1, Example 2].

Proposition 2.6. [1, Theorem 4.4] Let A⊂Ed. If reach(A0)≥Rthen Aadmits R-hull ˆ

Aand

A= (A0

R)0

Proposition 2.7. [1, Theorem 4.8] If A⊂E2is a connected subset of an open ball of radius R,

then A admits R-hull.

In the Appendix a more detailed proof is given. Let us also recall the following result:

Proposition 2.8. [1, Theorem 3.10], [12]) Let A⊂Edbe a closed set such that reach(A)≥R > 0. If

D⊂Edis a closed set such that for every a, b ∈D,h(a, b)⊂Dand A∩D6=∅, then reach(A∩D)≥R.

3 R-bodies

Let Rbe a ﬁxed positive real number. Bwill be any open ball of radius R.B(x) will be the open

ball of center x∈Edand radius R. Next deﬁnitions have been introduced in [10].

Deﬁnition 3.1. Let Abe a body, Awill be called an R-body if ∀y∈Ac,there exists an open ball B

in Ed(of radius R) satisfying y∈B⊂Ac.This is equivalent to say

Ac=∪{B:B∩A=∅};

that is

A=∩{Bc:B∩A=∅}.

Let us notice that for any r≥R, the body (B(x, r))cis an R-body.

Deﬁnition 3.2. Let E⊂Edbe a non empty set. The set

coR(E) := ∩{Bc:B∩E=∅}

will be called the R-hulloid of E. Let coR(E) = Edif there are no balls B⊂Ec.

Remark 3.3. In [10] the sets deﬁned in Deﬁnition 3.1 are called 2Rconvex sets and the sets deﬁned

in Deﬁnition 3.2 are called 2Rconvex hulls. On the other hand Meissner [9] and Valentine [15,

pp. 99-101] use the names of R-convex sets and R-convex hulls for diﬀerent families of sets. An

s-convex set is also deﬁned in [4, p. 42]. To avoid misunderstandings we decided to call R-bodies

and R-hulloids the sets deﬁned in Deﬁnition 3.1 and in Deﬁnition 3.2.

Remark 3.4. Let us notice that coR(E)is an R-body (by deﬁnition) and E⊂coR(E).Moreover A

is an R-body if and only if A=coR(A). The R-hulloid always exists.

Clearly every convex body Eis an R-body (for all positive R) and its convex hull co(E) = E

coincides with its R-hulloid.

Remark 3.5. It was noticed in [1, Corollary 4.7] and proved in [2, Proposition 1] that, when the

R-hull exists, it coincides with the R-hulloid. If Ahas reach greater or equal than R, then (see

remark 2.5) Ahas R-hull, which coincides with Aand with its R-hulloid, then Ais an R-body.

Proposition 3.6. Let Ebe a non empty set. The following facts have been proved in [10].

•acoR(E)=(E0

R)0

•bE−⊂coR(E);

•cLet E1⊂E2;then coR(E1)⊂coR(E2);

•dcoR(E1)∪coR(E2)⊂coR(E1∪E2);

•ecoR(coR(E)) = coR(E);

•fLet A(α), α ∈ A be R-bodies, then ∩α∈A A(α)is an R-body;

•gdiam E= diam coR(E);

•hIf Ais an R-body then Ais an r-body for 0< r < R;

•icoR(E)⊂co(E)for all R > 0.

Remark 3.7. Let Ebe a body. From cof Proposition 3.6 it follows that if Ais an R-body and

A⊃E, then A⊃coR(E)and coR(E)is the minimal R-body containing E.

Lemma 3.8. A point k∈coR(E)if and only if does not exist an open ball B(x, l)3kwith l≥R,

B(x, l)⊂EC.

Proof. As (B(x, l))cis an R-body, the set coR(E)∩(B(x, l))c⊃Ewould be an R-body strictly

included in coR(E), which is the minimal R-body containing E.

Lemma 3.9. Let Ebe a body. Then

(1) coR(E)⊂ER.

Moreover (ER)−may not be an R-body.

Proof. By 5of Proposition 2.2, (E0

R)0

R⊂ERand by aof Proposition 3.6, the inclusion (1) follows.

Let x0∈Ed, R < ρ < 2Rand let E= (B(x0, ρ))c.Then (ER)−is (B(x0, ρ −R))c,not an R-body.

Theorem 3.10. Let E⊂Edbe a body. Then

(2) coR(E) = ER∩∂(ER)0

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OnthecomplementsofunionofopenballsofxedradiusintheEuclideanspaceMarcoLonginetti*PaoloManselliAdrianaVenturiAbstractLetanR-bodybethecomplementoftheunionofopenballsofradiusRinEd.TheR-hulloidofaclosednotemptysetA,theminimalR-bodycontainingA,isinvestigated;ifAisthesetoftheverticesofasimplex,theR-hull...

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On the complements of union of open balls of xed radius in the Euclidean space Marco Longinetti

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