On the convex components of a set in $\mathbb{R}^n$

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli,"Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti,"Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008-2009) 71-78.

1. Introduction 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 k min (E) ≥ H n−1 (∂E) H n−1 (∂(co(E))) , (1.2) 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]. 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, §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 [4,9], 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 Figure 1. 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 1homogeneous 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 and (1.7) 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 the 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 r → 4h r + √ r 2 + 4h 2 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).

Main result.
The aim of the present paper is to improve the 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.
Definition 1.1 (Maximal sectional radius). Let n ≥ 2 and let E ⊂ R n be a compact set with non-empty interior. Given a direction ν ∈ R n , we let With the above definition in force, our main result reads as follows.

Theorem 1.2. Let n ≥ 2 and let E ⊂ R n be a compact set with non-empty interior.
Assume that there exist p ∈ N, α ∈ (0, 1) and β ∈ [0, 1] with the following properties. and 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 the 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 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 the 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 H 1 (∂(co(E))) < H 1 (∂E), 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 H n−1 (∂(co(E))) ≥ H n−1 (∂E), so that (1.2) only implies that k min (E) ≥ 1. Nevertheless, the 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, the 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 the 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 .

Examples
We dedicate the remaining part of the paper to give some explicit examples of compact sets E ⊂ R n , n ≥ 2, for which our main result applies. In each example, we will identify a point P ∈ ∂E and one convex component E j of E containing P and we will make a precise choice of parameters in order to satisfy the hypotheses of Theorem 1.2.
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 somē 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).
Since C is not convex, we must have that k min (C) ≥ 2. After all, it is evident that k min (C) = 3. Our argument will give such right value for a larger class of parameters l > h > 0 than the one provided in [6,Example 3.1]. First of all, notice that we do not deduce any further information from the result in [9]. Indeed, inequality (1.2) only yields since an elementary computation shows that H 1 (∂C) H 1 (∂(co(C))) = 2l + 2h l + 3h ∈ (1, 2) whenever l > h > 0. We now consider the point P ∈ ∂C as shown in Figure 2. For every decomposition of C into convex bodies, there exists a convex body E j containing P . Since E j is convex and contained in C, we must have that E j ⊂ H j , where H j is the halfspace such that ∂H j contains the face of C to which the point P belongs, see Figure 2.
Consequently, we must have where ν j ∈ S 1 is the inner unit normal of the half-space H j as in Figure 2. Now let l > 0 be fixed. In [6], it has been shown that, for any α ∈ (0, 1), p = 1 and h ≪ l, one has We now apply inequality (1.8) and Theorem 1.2 with In order to have both the claimed inequalities, it is sufficient to find h ∈ (0, l) such that that is, Up to some elementary algebraic computations, we need to find h ∈ (0, l) such that If we let h = tl for t ∈ (0, 1), then we just need to solve and we let the reader check that the above system of inequalities admits solutions.

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 the inequality (1.2) established in [9].   Figure 3. We can compute 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))) = 4l + 6h 4l + 5h + (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 halfspace such that ∂H j contains the face of L to which the point P belongs, see Figure 3. Consequently, we must have where ν j ∈ S 2 is the inner unit normal of the half-space H j as in Figure 3. We now let l > 0 be fixed. We apply Theorem 1.2 with Provided that we choose h ∈ (0, l) sufficiently small, we conclude that Since D is not convex, we must have that k min (D) ≥ 2, and a simple geometric argument allows to conclude that k min (D) = 3. From (1.2) we deduce that since an elementary computation shows that whenever l > 2h > 0. We now consider the point P ∈ ∂D as shown in Figure 4. For every decomposition of D into convex bodies, there exists a convex body E j containing P . Since E j is convex and contained in D, we must have that E j ⊂ H j , where H j is the halfspace such that ∂H j contains the face of D to which the point P belongs, see Figure 4. Consequently, we must have where ν j ∈ S 2 is the inner unit normal of the half-space H j as in Figure 4. We now let l > 0 be fixed. We apply Theorem 1.2 with Provided that we choose h ∈ 0, l 2 sufficiently small, we conclude that   Figure 5. We can compute H 2 (∂U) = 4hl + 10h 2 , H 2 (∂(co(U))) = 6hl + 4h 2 , diam(co(U)) = √ l 2 + 5h 2 .
Since U is not convex, we must have that k min (U) ≥ 2, and a simple geometric argument allows to conclude that k min (U) = 3. From (1.2) we deduce that since an elementary computation shows that H 2 (∂U) H 2 (∂(co(U))) = 4l + 10h 6l + 4h ∈ (0, 1) whenever l > 3h > 0. We now consider the points P, Q ∈ ∂U as shown in Figure 5. For every decomposition of U into convex bodies, there exists two convex bodies E j and E k containing P and Q respectively. Since the segment P Q is not contained in U, it follows that E j cannot contain Q. Since E j is convex and contained in U, we must have that E j ⊂ H j , where H j is the half-space such that ∂H j contains the face of U to which the point P belongs, see Figure 5. Consequently, we must have where ν j ∈ S 2 is the inner unit normal of the half-space H j as in Figure 5. By the symmetry of U, a similar argument can be used for the convex component E k containing Q. We now let l > 0 be fixed. We apply Theorem 1.2 with p = 2, α = l − h √ l 2 + 5h 2 , β = 1. Provided that we choose h ∈ (0, l 3 ) sufficiently small, we conclude that ∈ (1, 2).
The above computations prove that, in this case, although the lower bound given by (1.11) is strictly better than the one given by (1.2), the inequality (1.11) is not sharp. 3.
3. An example in R n . We conclude this section with Example 3.6 below, showing that for all n ≥ 3 our Theorem 1.2 provides an improvement of the inequality (1.2) established in [9]. In Example 3.6 we will need to apply the following result, whose elementary proof is detailed below for the reader's convenience. Proof. By definition, the set E n ⊂ R n satisfies H n (E n ) = ℓ n−2 H 2 (Q). for all n ≥ 2. The validity of (3.1) can thus be checked by induction, thanks to (3.2). Figure 6. The body L 2 ⊂ R 2 (on the left) and its convex hull (on the right).