A covering property of some classes of sets in R^{n}

Tamás Keleti

September 11, 2000

Abstract We prove that if B is a class of open bounded subsets of R^{n} satisfying a simple geometric condition then the following Besicovitch-type covering property is true. For any ε there exists an M such that from any subclass R ⊂ B one can select M subclasses of disjoint sets such that the selected sets cover at least the 1 − ε part of ∪R.

Thus we get sufficient geometric condition for the minimal density property and for the CV_{q} covering properties introduced in [2].

During the proof we also get a reverse isoperimetric inequality for the union of star-shaped sets.

1 The result

In this note we prove a covering result (Theorem 3) that can be interesting in itself but also has connection with the following recently defined notions [2]. (Throughout the paper |A| denotes the (Lebesgue) measure of A.)

Definition 1 Let B be a class of nonempty open bounded subsets of R^{n}.

The class B is said to have the minimal density property (MDP) if there exists a function ρ : R^{+} → R^{+} such that if H ⊂ R^{n} is measurable with finite measure, R ⊂ B covers H and the density of H in ∪R is d > 0 then one can find an R ∈ R in which the density of H is greater than ρ(d); that is,

\frac{|R ∩ H|}{|R|} > ρ (\frac{|H|}{| ∪ R|}) .

The class B is said to have the complete covering property V_{q} (CV_{q}) for a fixed 1 ≤ q ≤ ∞ if there exists a function C : R^{+} → R^{+} such that for any ε > 0 and R ⊂ B with | ∪ R| < ∞ we can find R_{1}, …, R_{m} ∈ R such that

(i) | ∪_{k=1}^{m} R_{k}| ≥ (1 − ε)| ∪ R| and (ii) ∥ ∑_{k=1}^{m} χ_{R_{k}}∥_{q} ≤ C(ε)| ∪ R|^{1/q},

where ∥.∥_{q} denotes the L_{q} norm and χ_{R} is the characteristic function of R.

Note that CV_{q} implies CV_{q′} if q > q′. It is proved in [2] that MDP and CV_{1} are equivalent and also that MDP implies that CV_{q} is equivalent with the classical (and weaker) covering property V_{q} for any 1 ≤ q < ∞. (The covering property V_{q} is defined in [1], where - among others - the authors proved that V_{q} is equivalent with the weak type (p, p) property of the maximal operator associated to B if 1 < q < ∞ and 1/p + 1/q = 1.)

Unfortunately, it is not easy to prove the minimal density property (even for the simplest classes, like the class of balls), which makes the applicability of this notion harder. (In [2] the MDP is proved only for the class of intervals of R^{n} (that is, n-dimensional axis-parallel rectangles)). It would be useful and interesting to have a weak sufficient geometric condition that guarantees the MDP. One can check (see [2] Example 4.7) that the class of sets in the plane that are the union of an open disc and an open sector with the same center and twice larger radius does not have the minimal density property . However, this is a regular class of sets (see Definition 7), which shows that the standard properties (regularity, V_{q} property (even for q = ∞), weak (1,1) property of the maximal operator, density property, differentiating properties) cannot guarantee the MDP. In this example the too “sharp” “thorn” is the obstacle of the MDP.

Our main result is the theorem below that shows that if the sets of B are “non-thorny” in the below defined sense then B has a much stronger property than the MDP or the CV_{q} properties: instead of (ii) of Definition 1, in this case, we have a better (Besicovitch type) control for the overlapping.

Definition 2 By a drop we mean the interior of the convex hull of a ball and a point (not contained in the ball). The angle of the drop is the angle between the line through the point and the center of the ball and any tangent line.

Let 0 < d < 1 and 0 < α < π/2. We say that a bounded open set H ⊂ R^{n} is (d, α)-non-thorny if H is the union of drops with angle at least α and diameter at least d · diamH.

Theorem 3 Let R be a family of (d, α)-non-thorny sets in R^{n} with bounded diameter. Then for any ε > 0 one can choose sets R_{1}, …, R_{m} ∈ R such that

(i)

| ∪_{k=1}^{m} R_{k}| ≥ (1 − ε)| ∪ R| and

(ii) the sequence R_{1}, …, R_{m} can be distributed in M families of disjoint sets, where M depends only on n, d, α and ε.

Remark 4 This covering property is similar to the Besicovitch property, the only difference is that, instead of all the centers, we cover a big part of the union. But, as the earlier mentioned example showed, in our case the Besicovitch property itself is not enough. However, we shall use the classical Besicovitch covering theorem (for balls) in the proof but we will also need estimate for the “edge” of the union of drops. This estimate will give us a reverse isoperimetric inequality for the union of star-shaped sets (Corollary 12), which can be interesting in itself.

Corollary 5 For any 0 < d < 1 and 0 < α < π/2, any class of (d, α)-non-thorny sets in R^{n} has the CV_{∞} property and consequently the CV_{q} property for any 1 ≤ q < ∞ and the minimal density property as well.

Therefore this non-thornity is a sufficient condition for the MDP but it is in fact too strong. However, as we shall see below, quite large and important classes satisfy it.

Definition 6 A set H ⊂ R^{n} is said to be star-shaped at x if \overline{xy} ⊂ H for every y ∈ H, where \overline{xy} denotes the closed segment between x and y.

The hub of H (hub(H)) is the set of all points at which H is star-shaped. Let r > 0. We say that H is r-star-shaped if hub(H) contains an open

ball with radius r · diamH.

Definition 7 A set H ⊂ R^{n} is r-regular if there exists a cube Q that contains H such that |H|/|Q| > r.

It is not hard to see (and probably well-known) that if H is a convex open r-regular set in R^{n} then H is r′-star-shaped, where r′ depends only on n and r. It is easy to see that any r-star-shaped set is (d, α)-non-thorny, where d and α depend only on r. Thus Theorem 3 has the following consequences:

Corollary 8 If R is a class of convex open r-regular sets or a class of rstar-shaped sets then for any ε > 0 one can select M subclasses of disjoint sets such that the selected sets cover the 1 − ε part of ∪R, where M depends only on n, r and ε.

Corollary 9 Any class of convex open r-regular sets or of r-star-shaped sets in R^{n} has the CV_{∞} property and consequently the CV_{q} property for any 1 ≤ q < ∞ and the minimal density property as well.

2 The proof of the result

Notation 10 Let x ∈ R^{n}, H ⊂ R^{n} and δ > 0. Let S(x, δ) denote the open ball with center x and radius δ. We denote the open neighborhood of H with radius δ by S(H, δ); that is,

S(H, δ) = ∪_{x∈H}S(x, δ).

We also introduce the δ-interior by the following definition:

int(H, δ) = {x : S(x, δ) ⊂ H}.

We denote the diameter of a set H by diam(H).

Lemma 11 Let H be a family of r-star-shaped sets in R^{n} with diameter D and let A = ∪H. Then for any δ ≤ D we have

|S(A, δ) \ A| ≤ C(n, r)\frac{δ}{D} |A|,  (1)

where C(n, r) depends only on n and r. (In fact, we can choose C(n, r) = ((1 + \frac{\sqrt{n}+1}{r})^{n} − 1)(\frac{\sqrt{n}+1}{r})^{n}.)

Proof. By homogeneity we can assume that D = 1.

For any H ∈ H there exists a ball S(O_{H}, r) ⊂ hub(H). Consider a cubic lattice with side \frac{2r}{\sqrt{n}+1} and for a lattice point P let S_{P} = S(P, \frac{r}{\sqrt{n}+1}). Let P_{H} be the nearest lattice point to O_{H}. Clearly, P_{H}O_{H} ≤ \sqrt{n}\frac{r}{\sqrt{n}+1}, so S_{P_{H}} ⊂ S(O_{H}, r) ⊂ hub(H). On the other hand, the balls S_{P} are disjoint.

For a lattice point P let

K_{P} = ∪{H ∈ H : S_{P} ⊂ hub(H)}.

Then A = ∪H = ∪_{P} K_{P} and for every lattice point P we have S_{P} ⊂ hub(K_{P}) and K_{P} ⊂ S(P, 1).

One can show (see e. g. [3] p. 286) that if K ⊂ S(P, 1) and S(P, a) ⊂ hub(K) then the magnification of K with center P and ratio 1 + \frac{δ}{a} contains S(K, δ). Then clearly

|S(K, δ) \ K| ≤ ((1 + \frac{δ}{a})^{n} − 1) |K| ≤ ((1 + \frac{δ}{a})^{n} − 1) |S(0, 1)|.

Therefore in our case we have

|S(K_{p}, δ) \ K_{p}| ≤ ((1 + \frac{δ}{r/(\sqrt{n} + 1)})^{n} − 1) |S(0, 1)|.

Thus, denoting by N the number of those lattice points P for which K_{P} is nonempty, we have

|S(A, δ) \ A| ≤ ∑_{P} |S(K_{p}, δ) \ K_{p}| ≤ N ((1 + \frac{δ(\sqrt{n} + 1)}{r})^{n} − 1) |S(0, 1)|.

On the other hand the balls S_{P} are disjoint subsets of A, hence

|A| ≥ ∑_{P} |S_{P} | = N (\frac{r}{\sqrt{n} + 1})^{n} |S(0, 1)|.

Therefore, using that δ ≤ D = 1, we get

\frac{|S(A, δ) \ A|}{δ|A|} ≤ (\frac{\sqrt{n} + 1}{r})^{n} \frac{((1 + ^{\frac{δ(\sqrt{n}+1)}{r}})^{n} − 1)}{δ}
≤ (\frac{\sqrt{n} + 1}{r})^{n} ((1 + \frac{\sqrt{n} + 1}{r})^{n} − 1) = C(n, r).

Corollary 12 If E is the union of r-star-shaped sets in R^{n} with diameter D then we have

\frac{Ã_{+}(E)}{|E|} ≤ \frac{C(n, r)}{D},

where Ã_{+}(E) denotes the upper outer surface area in the sense of Minkowski, that is

Ã_{+}(E) = lim sup_{δ→0+} \frac{|S(E, δ)| − |E|}{δ}.

Remark 13 If the diameters are not the same but between D_{1} and D_{2} then the same proof gives Ã_{+}(E)/|E| ≤ C(n, rD_{1}/D_{2})/D_{2}.

Remark 14 As a special case of Corollary 12, for example, we have that the ratio of the perimeter and the area of any finite union of (not necessary axis-parallel) unit squares is at most an absolute constant.

The author does not know the best constant. Is it 4?

Facts 15 Let D be a drop (see Definition 2) with angle 0 < α < π/2 and with diamD = d. Let E_{α} = \frac{1}{sin α} + 1 and δ < d/E_{α}. Then

1. the radius of the “ball part” of D is d/E_{α},

2. the set D is 1/E_{α}-star-shaped, 3. the set int(D, δ) (see Notation 10) is a drop with angle α and with

diameter d − E_{α}δ,

4. we have S(int(D, δ), E_{α}δ) ⊃ S(D, δ) and

5. for any 0 < d′ < d and 0 < α′ < α, D can be written as the union of drops with angle α′ and diameter d′.

Lemma 16 Let K be a family of (d, α)-non-thorny (see Definition 2) sets in R^{n} with diameter between Δ and 2Δ, let K = ∪K and let δ ≤ d/2E_{α}. Then one can choose sets K_{1}, …, K_{m} ∈ K such that

|S(K, δΔ) \ ∪_{k=1}^{m}K_{k}| ≤ Cδ|K|,  (2)

and the sequence K_{1}, …, K_{m} can be distributed in M(δ) families of disjoint sets, where C depends only on n, d and α and M(δ) depends only on n and δ.

Proof. By homogeneity, we can assume that Δ = 1.

Let D be the family of those drops with diameter d and angle α that are contained in at least one of the sets of K. Let B consist of the balls with radius δ contained in any drop of D. Put

D^{∗} = {int(D, δ) : D ∈ D} and K^{∗} = ∪D^{∗}.

Note that, by definition and Fact 15.5, K = ∪K = ∪D and that K^{∗} is covered by the centers of the balls of B. Thus, applying the classical covering theorem of Besicovitch, we get balls B_{1}, …, B_{m} ∈ B that cover K^{∗} but no point of R^{n} is covered more than C_{n} times. For k = 1, …, m let K_{k} be one of the sets of K that contain B_{k}. Then we have ∪K_{m} ⊃ K^{∗}.

We claim that every set K_{k} intersects at most C_{n}(4/δ)^{n} sets of the sequence K_{1}, …, K_{m} (including itself). Indeed, for a fixed k the sets K_{i}, that intersect K_{k}, are contained in a ball with radius 4 (since each set has diameter at most 2), but on the other hand, they contain balls with radius δ that cover each point at most C_{n} times, hence the number of sets that intersect K_{k} is at most C_{n}|S(0, 4)|/|S(0, δ)| = C_{n}(4/δ)^{n}.

Thus the sequence K_{1}, …, K_{m} can clearly be distributed in M(δ) = C_{n}(4/δ)^{n} families of disjoint sets: the greedy algorithm easily gives a proper distribution.

Now we prove (2). Using Fact 15.4 we get

S(K^{∗}, E_{α}δ) = ∪_{D∈D}S(int(D, δ), E_{α}δ) ⊃ ∪_{D∈D}S(D, δ) = S(∪D, δ)=S(K, δ).

Thus, using that ∪_{k=1}^{m}K_{k} ⊃ K^{∗}, we have

S(K, δ) \ ∪_{k=1}^{m}K_{k} ⊂ S(K^{∗}, E_{α}δ) \ K^{∗}.

According to Facts 15.2 and 15.3, D^{∗} consists of 1/E_{α}-star-shaped sets (in fact, drops) with diameter d − E_{α}δ. Therefore, using Lemma 11 for (D^{∗}, K^{∗}, 1/E_{α}, d − E_{α}δ, E_{α}δ) as (H, A, r, D, δ) and that δ ≤ d/2E_{α}, we get

|S(K^{∗}, E_{α}δ) \ K^{∗}| ≤ C(n, 1/E_{α})\frac{E_{α}δ}{d − E_{α}δ} |K^{∗}|
≤ C(n, 1/E_{α})\frac{E_{α}δ}{d/2} |K| = Cδ|K|,

where C = C(n, 1/E_{α})\frac{E_{α}}{d/2} depends only on n, d and α. This completes the proof of Lemma 16.

Proof of Theorem 3. Let N be a positive integer which will be defined later. By homogeneity, we can assume that each set of R has diameter at most 1/2. Let R_{k} (k = 1, 2, …) denote the family of sets of R with diameter between 1/2^{k+1} and 1/2^{k}, and let

H^{j} = R_{j} ∪ R_{N+j} ∪ R_{2N+j} ∪ … (j = 1, …, N).

Clearly R = H^{1} ∪ … ∪ H^{N}.

Fix j. Let K_{1} = R_{j}. If K_{1}, …, K_{l} is already defined then let K_{l+1} be the family of those sets of R_{Nl+j} which intersect no set of K_{1}, …, K_{l}. Then the diameters of the sets of K_{l} are between 1/2^{N(l−1)+j+1} and 1/2^{N(l−1)+j} (l = 1, 2, …). Let K_{l} = ∪K_{l} and δ_{l} = 1/2^{Nl+j}.

We claim that

∪H^{j} ⊂ ∪_{l=1}^{∞}S(K_{l}, δ_{l}).  (3)

Indeed, if x ∈ ∪H^{j} \ ∪_{l=1}^{∞}K_{l} then for an index i we have x ∈ R ∈ R_{Ni+j}. On the other hand R cannot be contained in K_{i+1}, so there must be an l ≤ i for which R intersects K_{l}. Since R ∈ R_{Ni+j} we have diamR ≤ 1/2^{Ni+j} ≤ δ_{l}. Thus x ∈ S(K_{l}, δ_{l}), which completes the proof of (3).

If we choose N such that 1/2^{N−1} ≤ d/2E_{α} then we can apply Lemma 16 for K = K_{l}, Δ = 1/2^{N(l−1)+j+1}, δ = 1/2^{N−1} to get K_{1}^{l}, …, K_{m_{l}}^{l} such that this sequence can be distributed in M(1/2^{N−1}) families of disjoint sets and

|S(K_{l}, δ_{l}) \ ∪_{i=1}^{m_{l}}K_{i}^{m_{l}}| ≤ C \frac{1}{2^{N−1}} |K_{l}|,

where C depends only on n, d and α.

Since the sets of K_{l} do not intersect the sets of K_{l′} (if l ≠ l′), the sets {K_{i}^{l} : l ∈ N, i = 1, …, m_{l}} can also be distributed in M(1/2^{N−1}) families of disjoint sets. On the other hand, we have

| ∪ H_{j} \ ∪_{i,l}K_{i}^{l}| ≤ | ∪_{l=1}^{∞} S(K_{l}, δ_{l}) \ ∪_{i,l}K_{i}^{l}| ≤ | ∪_{l=1}^{∞} (S(K_{l}, δ_{l}) \ ∪_{i=1}^{m_{l}}K_{i}^{l})|
≤ ∑_{l=1}^{∞} |(S(K_{l}, δ_{l}) \ ∪_{i=1}^{m_{l}}K_{i}^{l}| ≤ C \frac{1}{2^{N−1}} ∑_{l=1}^{∞} |K_{l}|
≤ \frac{C}{2^{N−1}} | ∪ H^{j}| ≤ \frac{C}{2^{N−1}} | ∪ R|.

Until this moment j was fixed. Now let R_{1}, R_{2}, … be the union of the families {K_{i}^{l}} we get for j = 1, …, N. Then these sets can be distributed in NM(1/2^{N−1}) families of disjoint sets and

| ∪ R \ ∪_{k}R_{k}| ≤ \frac{NC}{2^{N−1}} | ∪ R|.

Therefore, if N is an integer such that \frac{NC}{2^{N−1}} < ε and 1/2^{N−1} ≤ d/2E_{α} (depending only on n, d, α and ε) and M = NM(1/2^{N−1}) (depending also only on n, d, α and ε), then (i) and (ii) of Theorem 3 are satisfied if m is large enough.

Acknowledgment. The author would like to thank Professor D. Preiss for helpful discussions.

This research was, in part, supported by OTKA grant F 019468. Part of this research was done while the author was at the Mathematical Institute of the Hungarian Academy of Sciences. This note was completed while the author was visiting the University College London having a Royal Society/NATO Postdoctoral Fellowship award.

References

[1] A. Córdoba, R. Fefferman, A geometric proof for the strong maximal theorem, Ann. of Math. 102 (1975), 95-100.

[2] T. Keleti, Density and covering properties of intervals of R^{n}, submitted.

[3] S. Vrećica, A note on starshaped sets, Publ. Inst. Math. 29 (43) (1981), 283-288.

Department of Analysis, Eötvös Loránd University, Múzeum krt. 6-8, H-1088 Budapest, Hungary e-mail: elek@cs.elte.hu