Isoperimetric relations for inner parallel bodies

Inner parallel bodies of convex bodies have been object of recent studies with different flavors [6, 12, 13, 14, 15, 16, 17]. More classical existing literature on them (e.g. [1, 3, 4, 9, 10, 21]) along with its role in the proofs of fundamental results in the theory of convex bodies, make inner parallel bodies an essential object not only within classical Convex Geometry (see [22, Section 7.5]), but also in other related fields (see e.g. [7, 11, 20] and the references in [22, Note 3 for Section 3.1] and [6]).

In [6] and [16] the authors study the behavior of the isoperimetric quotient for the family of inner parallel bodies, and provide a lower bound for the perimeter of the inner parallel bodies of a convex body, respectively. However, in both articles they happen to make an erroneous use of the relation $K\subseteq K_{\lambda}+\left|K\right|K_{\lambda}^{*}$ between the inner parallel bodies $K_{\lambda}$ of a convex body, their form bodies $K_{\lambda}^{*}$ and the original convex body $K$ (see (2.4) and Section 2 for the proper definitions). This relation, which holds, for example, under technical properties of the boundary of the involved convex bodies (see (2.3)), is, however, not true without further conditions. To the best of the authors’ knowledge, a full characterization of the conditions under which the above inclusion holds is not known.

The purpose of this paper is two-sided. On the one hand, we describe the error contained in the two mentioned references, providing with examples proving these have to be adjusted with further hypotheses in order to hold. On the other hand, we provide alternative proofs to those results under suitable restrictions of the boundaries of the involved convex bodies, and further, we extend the results concerning inner parallel bodies in [6] to a more general setting.

The paper is organized as follows. In Section 2 we introduce the notions and basic results, which are needed throughout the paper. In Section 3 we analyze the problems in the proof of the main result in [6], providing an example where the used methods do not hold. In Section 4 we obtain new results concerning the behavior of the isoperimetric quotient and deficit under assumptions on the boundary of the involved convex bodies. Finally in Section 5 we point out an error -of the same spirit of the one found in [6]- in one of the proofs of [16] and discuss it.

Let $\mathcal{K}^{n}$ be the set of all convex bodies, i.e., nonempty compact convex subsets of the Euclidean space $\mathbb{R}^{n}$, and let $\mathcal{K}^{n}_{n}$ the subset of convex bodies having interior points. A convex body $K$ is called regular if all its boundary points are regular, i.e., the supporting hyperplane to $K$ at any boundary point is unique. Let $\mathrm{B}_{n}$ be the $n$-dimensional Euclidean unit ball and $\mathbb{S}^{n-1}$ the corresponding unit sphere. The volume of a measurable set $M\subset\mathbb{R}^{n}$, i.e., its $n$-dimensional Lebesgue measure, is denoted by $\mathrm{vol}(M)$, and the measure of its boundary, i.e., its surface area (also called perimeter), is represented by $\mathrm{S}(M)$. Furthermore, the closure of $M$ is denoted by $\mathop{\mathrm{cl}}\nolimits M$. For $K\in\mathcal{K}^{n}$ and $u\in\mathbb{S}^{n-1}$, $h(K,u)=\sup\bigl{\{}\langle x,u\rangle:x\in K\bigr{\}}$ stands for the support function of $K$ (see e.g. [22, Section 1.7]).

The vectorial or Minkowski addition of two sets $K,L\subset\mathbb{R}^{n}$ is given by

whereas the Minkowski difference of $K,L\subset\mathbb{R}^{n}$ is given by

We notice that $(K\sim L)+L\subseteq K$, and the inequality may be strict.

Let $K\in\mathcal{K}^{n}$ and $E\in\mathcal{K}^{n}_{n}$. The inradius $\mathrm{r}(K;E)$ of $K$ relative to $E$ is the radius of one of the largest dilations of $E$ which fits inside $K$, i.e.,

For $-\mathrm{r}(K;E)\leq\lambda\leq 0$ the inner parallel body of $K$ at distance $\left|\lambda\right|$ is the Minkowski difference of $K$ and $\left|\lambda\right|E$, i.e.,

Notice that if $E=B_{n}$, then $K_{-\mathrm{r}(K;\mathrm{B}_{n})}$ is the set of incenters of $K$, which is usually called the kernel of $K$, and its dimension is strictly less than $n$ (see [2, p. 59]). Equivalently (see [22, Section 3.1]), the inner parallel body $K_{\lambda}$ of $K$, $-\mathrm{r}(K;E)\leq\lambda\leq 0$, can be defined using the support functions of $K$ and $E$ as

A vector $u\in\mathbb{S}^{n-1}$ is a $0$-extreme normal vector (or just extreme vector) of $K$ if it cannot be written as a linear combination of two linearly independent normal vectors at one and the same boundary point of $K$. We denote by $\mathcal{U}(K)$ the set of $0$-extreme normal vectors of $K$, which play a key role in the study of convex bodies. For instance, $\mathcal{U}(K)$ is the smallest set one can use so that

(see e.g. [22, Corollary 1.4.5 or page 386]), and thus, the inner parallel bodies of $K$ can be expressed as (cf. (2.1))

for $-\mathrm{r}(K;E)\leq\lambda\leq 0$.

The (relative) form body of a convex body $K\in\mathcal{K}^{n}_{n}$ with respect to $E\in\mathcal{K}^{n}_{n}$, denoted by $K^{*}$, is defined as (see e.g. [3])

We notice that $K^{*}$ strongly depends on the body $E$. Nevertheless, and for the sake of simplicity, we omit $E$ in the notation.

The form body of $K\in\mathcal{K}^{n}_{n}$ (with respect to an arbitrary $E\in\mathcal{K}^{n}_{n})$ is always a tangential body of $E$. We recall that a convex body $K\in\mathcal{K}^{n}$ containing a convex body $E\in\mathcal{K}^{n}$, is called a tangential body of $E$, if through each boundary point of $K$ there exists a support hyperplane to $K$ that also supports $E$. We notice that if $K$ is a tangential body of $E$, then $\mathrm{r}(K;E)=1$.

There is also a very close connection between inner parallel bodies and tangential bodies. The next result enlighten it.

In the following, we collect some standard properties of inner parallel bodies, form bodies and extreme vectors, together with other relations through the Minkowski sum, which will be needed later on. There exist further relations, in a stronger form, through the so-called Riemann-Minkowski integral, for which we refer to [5] and [21, Lemma 3.2].

Let $K,E\in\mathcal{K}^{n}_{n}$. From now on we will write $K_{\lambda}^{*}=(K_{\lambda})^{*}$ to denote the form body of the inner parallel body of $K$ at distance $\left|\lambda\right|$, $-\mathrm{r}(K;E)<\lambda\leq 0$. The following counterpart of the relations contained in Lemma 2.3 (v), can be found in [21, Corollary to Lemma 4.8] (see also Lemma 2.3 (ii)).

Condition (2.3) deserves further observations. On the one hand, it is similar to the identity $\mathcal{U}(K_{\lambda}+K^{*})=\mathcal{U}(K^{*})$, which is a direct consequence of the relation $\mathcal{U}(K_{\lambda}+K^{*})=\mathcal{U}(K)$ needed in [14, Theorem 2.2] (see Remark 2.4), together with Lemma 2.3 (ii) and (iii). However, since examples of convex bodies for which $\mathcal{U}(K_{\lambda})\subset\mathcal{U}(K)$ (strictly) are easily constructed, both conditions are different. On the other hand, Lemma 2.3 (iii) yields $\mathcal{U}(K_{\lambda}^{*})\subseteq\mathcal{U}(K_{\lambda}+K_{\lambda}^{*})$; however, the inclusion may be strict (see Section 3).

Let $K\in\mathcal{K}^{n}$ be a convex body. In [6] the authors study the isoperimetric quotient $I(K_{-\lambda}):=\mathrm{vol}(K_{-\lambda})/\mathrm{S}(K_{-\lambda})^{n/(n-1)}$ of the family of inner parallel bodies $K_{-\lambda}$, $0\leq\lambda<\mathrm{r}(K;\mathrm{B}_{n})$, when $E=\mathrm{B}_{n}$, and analyze the behavior of the function $I(\lambda)=I(K_{-\lambda})$: in Theorem 1 they prove that the isoperimetric quotient function $I(\lambda)$ is non-increasing in $0\leq\lambda<\mathrm{r}(K;\mathrm{B}_{n})$ for all convex bodies. However, as we mentioned in the introduction, the proof of this result is erroneous.

The main idea of the proof is to bound from below the quotient defining $I(\lambda)$. To this end, the numerator $\mathrm{vol}(K_{\lambda})$ is bounded using Lemma 2.3 (iv) and the property $(K_{-\lambda})_{-\mu}=K_{-\lambda-\mu}$ for $0\leq\lambda,\mu\leq\lambda+\mu<\mathrm{r}(K;\mathrm{B}_{n})$ (see [22, (3.17)]). More precisely, and following the notation in [6], for $0\leq t\leq t_{0}<\mathrm{r}(K;\mathrm{B}_{n})$,

In order to bound the denominator $\mathrm{S}(K_{-t})$, the authors make use of the monotonicity of the surface area applied to the content (2.4), namely,

for $0\leq t\leq t_{0}<\mathrm{r}(K;\mathrm{B}_{n})$. However, this inclusion is not true without further conditions (as, for instance, the equality $\mathcal{U}(K_{-t_{0}}^{*})=\mathcal{U}(K_{-t_{0}}+K_{-t_{0}}^{*})$, see Proposition 2.5). Indeed, in the next, we prove that the content $K\subseteq K_{\lambda}+\left|\lambda\right|K_{\lambda}^{*}$, $-\mathrm{r}(K;E)<\lambda\leq 0$, is not valid in its full generality.

For $K\in\mathcal{K}^{n}_{n}$ and $\mu\geq 0$, we consider the following convex body:

This construction appeared already in [16, 17, 21]. Indeed, in [21] the following result was proven.

We will also need the following additional result.

We are now in a position to prove the announced non-validity of the inclusion (2.4) without further assumptions. Next result will be proven when the body $E=\mathrm{B}_{n}$, although it holds true for any regular $E\in\mathcal{K}^{n}_{n}$.

We start pointing out that unlike the authors in [6], we are going to consider the family of inner parallel bodies defined in the range $-\mathrm{r}(K;E)\leq\lambda\leq 0$, which will reverse the behavior of the isoperimetric quotient function (in the original paper [6] the range is $0\leq\lambda\leq\mathrm{r}(K;\mathrm{B}_{n})$ what makes the behavior of $I(\lambda)$ non-increasing). Furthermore, since we will also work with the isoperimetric deficit, we will consider the isoperimetric quotient in the usual way, namely, $\mathrm{S}(K_{\lambda})^{n}/\mathrm{vol}(K_{\lambda})^{n-1}$, in order to compare the behavior in both cases.

In this section we are going to obtain new results concerning the behavior of the isoperimetric quotient (and also of the isoperimetric deficit) under assumptions on the boundary of the involved convex bodies. The first condition we can impose is, actually, the boundary condition necessary to validate the proof of Theorem 1 in [6], namely:

The proof of this result is exactly the proof of [6, Theorem 1], where the use of (2.4) is justified by assuming (2.3).

Next we prove that under different conditions to (2.3), the isoperimetric quotient function is also non-decreasing. In fact, we will get a more general result for all the quermassintegrals of a convex body $K$ (relative to an arbitrary $E\in\mathcal{K}^{n}_{n}$), which we define next.

Given $K\in\mathcal{K}^{n}$ and $E\in\mathcal{K}^{n}_{n}$, the so-called relative Steiner formula states that the volume of the Minkowski addition $K+\mu E$, $\mu\geq 0$, is a polynomial of degree $n$ in $\mu$,

The coefficients $\mathrm{W}_{i}(K;E)$ are called (relative) quermassintegrals of $K$, and they are just a special case of the more general mixed volumes, for which we refer to [22, s. 5.1]. In particular, we have $\mathrm{W}_{0}(K;E)=\mathrm{vol}(K)$ and $\mathrm{W}_{n}(K;E)=\mathrm{vol}(E)$. Moreover, if $E=\mathrm{B}_{n}$, the polynomial in the right hand side becomes the classical Steiner polynomial, see [23], and $n\mathrm{W}_{1}(K;\mathrm{B}_{n})=\mathrm{S}(K)$ is the usual surface area of $K$.

Let $\mathrm{W}_{i}(\lambda):=\mathrm{W}_{i}(K_{\lambda};E)$ for $-\mathrm{r}(K;E)\leq\lambda\leq 0$. From the concavity of the family of inner parallel bodies (see [22, Lemma 3.1.13]) and the general Brunn-Minkowski theorem for relative quermassintegrals (see e.g. [22, Theorem 7.4.5]), it is obtained that

for $i=0,\dots,n-1$ and for $-\mathrm{r}(K;E)\leq\lambda\leq 0$. Here ${}^{\prime}\mathrm{W}_{i}$ and $\mathrm{W}_{i}^{\prime}$ denote, respectively, the left and right derivatives of the function $\mathrm{W}_{i}(\lambda)$, and for $\lambda=-\mathrm{r}(K;E)$ (respectively, $\lambda=0$) only the right (left) derivative is considered ($\mathrm{W}_{i}^{\prime}$ will also denote the full derivative of $\mathrm{W}_{i}$ when the function is differentiable). In [12] the following definition was introduced.

Notice that the class $\mathcal{R}_{p}$ depends on the fixed convex body $E$. Nevertheless, and for the sake of simplicity, we will also omit $E$ in the notation.

Since the volume is always differentiable with respect to $\lambda$ and $\mathrm{vol}^{\prime}(\lambda)=n\mathrm{W}_{1}(\lambda)$ (see e.g. [1, 18]), the class $\mathcal{R}_{0}$ consists of all convex bodies, i.e., $\mathcal{R}_{0}=\mathcal{K}^{n}$. From the definition we get $\mathcal{R}_{p}\supset\mathcal{R}_{p+1}$, $p=0,\dots,n-2$, and all these inclusions are strict (particular tangential bodies show it; see [12]). The problem of determining the convex bodies belonging to the class $\mathcal{R}_{p}$ was studied by Bol [1] and Hadwiger [9] in the 3-dimensional case when $E=\mathrm{B}_{n}$. In [12] and [15] the general classes $\mathcal{R}_{n-1}$ and $\mathcal{R}_{n-2}$, respectively, were characterized. The cases $p=1,\dots,n-3$ remains open.

Finally, we recall the following inequalities for quermassintegrals, which can be deduced from the well-known Aleksandrov-Fenchel inequalities for mixed volumes (see e.g. [22, Sections 7.3 and 7.4]). They motivate and are also needed to prove our results. Let $K\in\mathcal{K}^{n}$ and $E\in\mathcal{K}^{n}_{n}$. Then

and

We notice that the last inequality, for $E=\mathrm{B}_{n}$ and $i=0$, $j=1$, yields the well-known isoperimetric inequality $\mathrm{S}(K)^{n}\geq n^{n}\mathrm{vol}(\mathrm{B}_{n})\mathrm{vol}(K)^{n-1}$.

In [16, Theorem 1.2] the author provides a lower bound for the surface area of the inner parallel bodies of a convex body $K$ (with respect to $\mathrm{B}_{n}$); following our notation, it is shown that