On the convex components of a set in ℝ ⁿ

1.1 Convex components

Let n ≥ 2 . Let us consider a compact set E ⊂ ℝ n with non-empty interior and a decomposition of the form

where k ∈ ℕ and E 1 , … , E k are the convex components of E, i.e., compact and convex sets with non-empty interior. Since, in general, such a decomposition is obviously not unique, it is interesting to give a lower bound on the minimal number k min ⁢ ( E ) ∈ ℕ of the convex components of E. By definition, k min ⁢ ( E ) = 1 if and only if E is a convex body. Moreover, we can note that k min ⁢ ( E ) ≥ c ⁢ ( E ) , where c ⁢ ( E ) ∈ ℕ is the number of connected components of E. Indeed, any convex component of E must lay inside some connected component of E. Therefore, without loss of generality, in the following we will always assume that E is a connected set.

The first lower bound on the minimal number of convex components was given in [9, Theorem 1.1], where the authors proved that

where ⌈ x ⌉ ∈ ℤ denotes the upper integer part of x ∈ ℝ . Here and in the following, for all s ≥ 0 we let ℋ s be the s-dimensional Hausdorff measure (in particular, ℋ 0 is the counting measure). Moreover, we let ∂ ⁡ E be the boundary of E and co ⁡ ( E ) be the convex hull of E. Note that, since E admits at least one decomposition as in (1.1), ℋ n - 1 ⁢ ( ∂ ⁡ E ) and ℋ n - 1 ⁢ ( ∂ ⁡ ( co ⁡ ( E ) ) ) are two finite and strictly positive real numbers, see [9], so that the right-hand side in (1.2) is well defined.

In the subsequent paper [6], the bound in (1.2) has been improved in the case n = 2 in the sense explained in Section 1.3 for a class of compact sets E ⊂ ℝ 2 . In the same spirit, our aim is to provide a refined bound of the number k min ⁢ ( E ) in any dimension n ≥ 2 . We stress that the estimate we are going to obtain also improves the result in [6].

1.2 Monotonicity of perimeter

The proof of (1.2) is based on the following monotonicity property of the perimeter: if A ⊂ B ⊂ ℝ n are two convex bodies, then

Inequality (1.3) is well known since the ancient Greek (Archimedes himself took it as a postulate in his work on the sphere and the cylinder, see [1, p. 36]) and can be proved in many different ways, for example by exploiting either the Cauchy formula for the area surface or the monotonicity property of mixed volumes, [2, Section 7], by using the Lipschitz property of the projection on a convex closed set, [3, Lemma 2.4], or finally by observing that the perimeter is decreased under intersection with half-spaces, [10, Exercise 15.13]. Actually, a deep inspection of the proof given in [3] shows that the convexity of B is not needed.

Anyway, in [9], a quantitative improvement of formula (1.3) has been obtained if A and B are both convex bodies. Moreover, lower bounds for the perimeter deficit

with respect to the Hausdorff distance of A and B have been established for n = 2 in [9, 4], for n = 3 in [5] and finally for all n ≥ 2 in [11].

In particular, if A ⊂ B are two convex bodies in ℝ n , with n ≥ 2 , then

where ω n = π n / 2 Γ ⁢ ( n 2 + 1 ) denotes the volume of the unit ball in ℝ n , h = h ⁢ ( A , B ) is the Hausdorff distance of A and B and

with a ∈ A and b ∈ B such that | a - b | = h ⁢ ( A , B ) , see [11, Corollary 1.2] and Figure 1.

Figure 1

The setting of the estimate (1.4) (on the left) with an example of equality (on the right).

Actually, the main result of [11] provides a quantitative lower bound for the more general deficit

where P Φ stands for the anisotropic (Wulff) perimeter associated to the positively 1-homogeneous convex function Φ : ℝ n → [ 0 , + ∞ ) .

We conclude this subsection by underlying that the quantitative estimates of the perimeter deficit δ ⁢ ( B , A ) obtained in [4, 5, 11] are sharp in the sense that they hold as equalities in some cases, see Figure 1.

1.3 Improvement of (1.2) in the planar case

Taking advantage of the quantitative estimate (1.4) in the planar case proved in [5], in the more recent paper [6] the authors were able to improve the lower bound (1.2) for n = 2 for a class of compact sets E ⊂ ℝ 2 (see also [7]). Precisely, if for a bounded closed ∅ ≠ E ⊂ ℝ 2 one can find q ∈ ℕ 0 , p ∈ ℕ and α ∈ ( 0 , 1 ) such that any decomposition of the form (1.1) admits p convex components E i 1 , … , E i p such that

and

then

Inequality (1.7) is sharp, in the sense that it holds as an equality in some cases. Moreover, it improves the previous lower bound (1.2) in the case n = 2 . Indeed, in [6] the authors exhibited an example for which (1.2) gives a strict inequality while, on the contrary, (1.7) yields an equality.

The idea behind inequality (1.7) essentially relies on two ingredients. On the one hand, the use of the refined estimate of the deficit obtained in [4] in place of the monotonicity property of the perimeter (1.3). On the other hand, the idea of assuming (1.5) for a finite number p of the components, according to the observation that some planar sets E ⊂ ℝ 2 have some convex components whose Hausdorff distance from the convex hull co ⁡ ( E ) is comparable to the diameter of co ⁡ ( E ) itself, independently of the chosen decomposition.

By a careful inspection of the proof of (1.7), one realizes that

and since the function

is monotone for r > 0 , the assumption (1.5) yields

which is precisely (1.7), according to the best possible choice of q ∈ ℕ 0 in (1.6).

1.4 Main result

The aim of the present paper is to improve inequality (1.2) for all n ≥ 2 exploiting the quantitative monotonicity of the perimeter (1.4) proved in [11], thus generalizing inequality (1.8) to higher dimensions. Before stating our main result, we need to introduce the following notation.

With the above definition in force, our main result reads as follows.

1.5 Comments

First of all, let us remark that inequality (1.11) improves the previous lower bound (1.2). Indeed, inequality (1.11) clearly reduces to the lower bound (1.2) as soon as one drops the additional assumptions on each of all possible decompositions of the form (1.1). Moreover, inequality (1.11) holds as an equality in some cases for which (1.2) gives a strict inequality only. We will give some explicit examples in Section 3 below.

Concerning the statement of Theorem 1.2, it is worth noting that the assumption (1.9) corresponds to (1.5), while the additional assumption (1.10) comes into play for n ≥ 3 only.

In fact, if we take n = 2 in Theorem 1.2, then inequality (1.11) becomes

(as it is customary, we use the convention 0 0 = 1 ) and the parameter β ∈ [ 0 , 1 ] provided by (1.10) plays no role in the final estimate (1.12). Consequently, the additional assumption in (1.10) can be dropped and one just needs to choose the closed half-plane H i j ⊂ ℝ 2 in such a way that

which is always possible by the definition of the Hausdorff distance and the convexity of each component E i j .

Concerning the higher-dimensional case n ≥ 3 , a control like the one in (1.10) seems reasonable to be assumed. Indeed, as one may realize by looking at inequality (1.2), the set E ⊂ ℝ n may have a convex component very lengthened in one specific direction ν ∈ 𝕊 n - 1 which does not give a substantial contribution to the total perimeter of E but, nevertheless, that strongly affects the total perimeter of the convex hull co ⁡ ( E ) .

In addition, we observe that the effectiveness of the lower bound (1.2) drastically changes when passing from the planar case n = 2 to the non-planar case n ≥ 3 . Indeed, if E ⊂ ℝ 2 is a non-convex connected compact set admitting at least one decomposition like (1.1), then

correctly implying that k min ⁢ ( E ) ≥ 2 . As a matter of fact, in the planar case n = 2 , the examples given in [6] provide the precise value of k min ⁢ ( E ) for q ≥ 2 , since if q = 1 both inequalities (1.2) and (1.7) allow to conclude that k min ⁢ ( E ) ≥ 2 only. However, as we are going to show with some examples in Section 3 below, there are non-convex connected compact sets E ⊂ ℝ n , with n ≥ 3 , such that

so that (1.2) only implies that k min ⁢ ( E ) ≥ 1 . Nevertheless, inequality (1.11) given by Theorem 1.2 allows us to recover the correct value of k min ⁢ ( E ) in these examples.

Moreover, let us observe that, in the planar case n = 2 , one can trivially bound

so that inequality (1.12) gives back

that is the estimate in (1.8). Actually, because of the fact that the upper bound (1.13) can be too rough in general, inequality (1.12) given by our Theorem 1.2 is more precise than the one in (1.8), as we are going to show in Example 3.1 below.

Last but not least, we remark that both the lower bounds provided by estimates (1.2) and (1.11) are not stable under small modifications of the compact set E ⊂ ℝ n , n ≥ 2 . In fact, the value of k min ⁢ ( E ) may be changed without substantially altering neither the perimeters of E and of its convex hull co ⁡ ( E ) , nor all the other geometrical quantities involved in (1.11), for example by gluing some additional tiny convex components to the original set E.

1.6 Organization of the paper

The rest of the paper is organized as follows.

In Section 2 we detail the proof of our main result, Theorem 1.2. Our approach essentially follows the strategy of [6], up to some minor modifications needed in order to exploit the quantitative estimate (1.4) in conjunction with the notion of maximal radius introduced in Definition 1.1.

In Section 3 we provide some examples proving the effectiveness of our main result with respect to either the general inequality (1.2) or its improvement (1.8) in the planar case, as already observed, due to the fact that ρ ν ⁢ ( co ⁡ ( E ) ) ≤ diam ⁡ ( co ⁡ ( E ) ) 2 for all ν ∈ 𝕊 1 .

3.1 An example in ℝ 2

We begin with the following example in ℝ 2 showing that our Theorem 1.2 in the planar formulation (1.12), at least in some cases, provides a strictly better estimate than the one in (1.8) previously established in [6]. This example is based on the set C ⊂ ℝ 2 shown in Figure 2, which was already considered in [9, Example 2.1] and in [6, Example 3.1]. The set C depends on two parameters l > h > 0 . In [6, Example 3.1], to make the construction work, it was necessary to assume that h ∈ ( 0 , ε ) for some ε ∈ ( 0 , l ) sufficiently small. In our situation, thanks to the refined inequality (1.8), our choice of the parameter h is less restrictive, i.e., we are going to choose h ∈ ( 0 , ε ¯ ) for some ε ¯ ∈ ( ε , l ) . As matter of fact, when h ∈ ( ε , ε ¯ ) , our inequality (1.8) gives the correct value k min ⁢ ( C ) = 3 , while inequality (1.7) gives the lower bound k min ⁢ ( C ) ≥ 2 only.

Figure 2

The set C ⊂ ℝ 2 (on the left) and its convex hull (on the right).

3.2 Some examples in ℝ 3

We now give some examples in ℝ 3 showing that for n = 3 our Theorem 1.2 provides an improvement of inequality (1.2) established in [9].

Example 3.2 (The set L ⊂ R 3 ).

Let l > h > 0 and consider the set L ⊂ ℝ 3 in Figure 3. We can compute

ℋ 2 ⁢ ( ∂ ⁡ L ) = 4 ⁢ h ⁢ l + 6 ⁢ h 2 ,

ℋ 2 ⁢ ( ∂ ⁡ ( co ⁡ ( L ) ) ) = 4 ⁢ h ⁢ l + 5 ⁢ h 2 + h ⁢ ( l - h ) 2 + h 2 ,

diam ⁡ ( co ⁡ ( L ) ) = l 2 + 5 ⁢ h 2 .

Since L is not convex, we must have that k min ⁢ ( L ) ≥ 2 , and a simple geometric argument allows to conclude that k min ⁢ ( L ) = 2 . From (1.2) we deduce that

k min ⁢ ( L ) ≥ ⌈ ℋ 2 ⁢ ( ∂ ⁡ L ) ℋ 2 ⁢ ( ∂ ⁡ ( co ⁡ ( L ) ) ) ⌉ = 1 ,

since an elementary computation shows that

ℋ 2 ⁢ ( ∂ ⁡ L ) ℋ 2 ⁢ ( ∂ ⁡ ( co ⁡ ( L ) ) ) = 4 ⁢ l + 6 ⁢ h 4 ⁢ l + 5 ⁢ h + ( l - h ) 2 + h 2 ∈ ( 0 , 1 )

whenever l > h > 0 . We now consider the point P ∈ ∂ ⁡ L as shown in Figure 3. For every decomposition of L into convex bodies, there exists a convex body E j containing P. Since E j is convex and contained in L, we must have that E j ⊂ H j , where H j is the half-space such that ∂ ⁡ H j contains the face of L to which the point P belongs, see Figure 3. Consequently, we must have

h ⁢ ( co ⁡ ( L ) ∩ H j , co ⁡ ( L ) ) = l - h , ℋ 2 ⁢ ( co ⁡ ( L ) ∩ ∂ ⁡ H j ) = 2 ⁢ h 2 , ρ ν j ⁢ ( co ⁡ ( L ) ) = 2 ⁢ h 2 π ,

where ν j ∈ 𝕊 2 is the inner unit normal of the half-space H j as in Figure 3.

Figure 3

The set L ⊂ ℝ 3 (on the left) and its convex hull (on the right).

We now let l > 0 be fixed. We apply Theorem 1.2 with

p = 1 , α = l - h l 2 + 5 ⁢ h 2 , β = 1 .

Provided that we choose h ∈ ( 0 , l ) sufficiently small, we conclude that

k min ⁢ ( L ) ≥ ⌈ 1 4 ⁢ h ⁢ l + 5 ⁢ h 2 + h ⁢ ( l - h ) 2 + h 2 ⁢ ( 4 ⁢ h ⁢ l + 6 ⁢ h 2 + π ⁢ ( l - h l 2 + 5 ⁢ h 2 ) 2 ⁢ 2 ⁢ h 2 π ⁢ ( l 2 + 5 ⁢ h 2 ) 2 2 ⁢ h 2 π + 2 ⁢ h 2 π + ( l - h l 2 + 5 ⁢ h 2 ) 2 ⁢ ( l 2 + 5 ⁢ h 2 ) 2 ) ⌉

= ⌈ 1 4 ⁢ l + 5 ⁢ h + ( l - h ) 2 + h 2 ⁢ ( 4 ⁢ l + 6 ⁢ h + 2 ⁢ π ⁢ ( l - h ) 2 2 ⁢ h 2 π + 2 ⁢ h 2 π + ( l - h ) 2 ) ⌉ = 2 ,

since

lim h → 0 + ⁡ 4 ⁢ l + 6 ⁢ h + 2 ⁢ π ⁢ ( l - h ) 2 2 ⁢ h 2 π + 2 ⁢ h 2 π + ( l - h ) 2 4 ⁢ l + 5 ⁢ h + ( l - h ) 2 + h 2 = 4 + 2 ⁢ π 5 ∈ ( 1 , 2 ) .