On a reverse isoperimetric inequality for relative outer parallel bodies
Eugenia Saor\'in G\'omez, Jes\'us Yepes Nicol\'as
TL;DR
This paper establishes a reverse isoperimetric inequality for relative outer parallel bodies with respect to a convex body, providing conditions for equality and exploring related inequalities through convexity properties.
Contribution
It introduces a novel reverse isoperimetric inequality for relative outer parallel bodies and analyzes equality conditions and related inequalities using quermassintegrals.
Findings
Reverse isoperimetric inequality proved for relative outer parallel bodies.
Equality conditions characterized for the inequality.
Additional inequalities derived from convexity of quermassintegrals.
Abstract
We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body , along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.
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Taxonomy
TopicsPoint processes and geometric inequalities · Pharmacological Effects of Medicinal Plants · Morphological variations and asymmetry
On a reverse isoperimetric inequality for relative outer parallel bodies
Eugenia Saorín Gómez
ALTA Institute for Algebra, Geometry, Topology and their Applications, Universität Bremen, D-28359 Bremen, Germany
and
Jesús Yepes Nicolás
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain
Abstract.
We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body , along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.
Key words and phrases:
Steiner formula, Minkowski sum, relative quermassintegrals, convex sequences
2010 Mathematics Subject Classification:
52A20, 52A40
Both authors are supported by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 19901/GERM/15. Second author is supported by MINECO/FEDER project MTM2015-65430-P and MICINN/FEDER project PGC2018-097046-B-I00.
1. Introduction
Let denote the set of all convex bodies in , i.e., the set of all non-empty compact convex subsets of . For two convex bodies and a non-negative real number , the volume (i.e., the Lebesgue measure) of the Minkowski sum is expressed as a polynomial of degree at most in , and it is written as
[TABLE]
This expression is called Minkowski-Steiner formula or relative Steiner formula of . The coefficients are the relative quermassintegrals of , and they are a special case of the more general defined mixed volumes for which we refer to [11, Chapter 6] and [16, Section 5.1]. In particular, we have , , for and, if has dimension , if and only if . Moreover, is the (relative) Minkowski content of , which will be denoted by . Quermassintegrals admit also a Steiner formula, namely
[TABLE]
for any .
A finite sequence of real numbers is called concave (see e.g. [16, Section 7.4]) if
[TABLE]
or equivalently, if
[TABLE]
Moreover, a sequence is said to be convex if it satisfies the reversed inequality to (1.3), i.e., , for .
As a consequence of this, a concave sequence satisfies the following inequality [16, (7.59)]:
[TABLE]
for any . In the latter, there is equality if and only if
[TABLE]
When the sequence contains only positive numbers, it is called log-concave if the logarithm of the sequence, namely is concave, which is equivalent to the following inequalities [16, Section 7.4]:
[TABLE]
For a convex body the fundamental inequalities
[TABLE]
which are consequences of the more general Aleksandrov-Fenchel inequalities for mixed volumes (see e.g. [16, Section 7.3]) do yield that the quermassintegrals of (relative to ) constitute a log-concave sequence, i.e., they satisfy (1.5). Furthermore, when dealing with -dimensional convex bodies with a common projection onto a hyperplane, then the sequence of relative quermassintegrals is also concave and thus it satisfies (1.4) (see [16, Theorem 7.7.1 and (7.190)]).
Here we show that, moreover, the reverse form of (1.4) holds when assuming that is a summand of :
Theorem 1.1**.**
Let with and . Then, for every ,
[TABLE]
Equality holds if and only if .
By taking , and in Theorem 1.1 we get the following result:
Corollary 1.1**.**
Let with and . Then
[TABLE]
Equality holds if and only if .
The latter inequality can be regarded as a reverse form of the well-known (relative) isoperimetric inequality for -dimensional convex bodies (also referred to in the literature as the Minkowski first inequality, see e.g. [16, Theorem 7.2.1]):
[TABLE]
and equality holds if and only if (up to translations) for some . Another result in this regard is the celebrated reverse isoperimetric inequality, due to Ball [1] (with equality conditions later supplied by Barthe [3]), in the classical setting (see also [2] and the references therein).
The case E=\bigl{(}1/\lambda\bigr{)}B_{n} of the Euclidean ball of radius (for ) in both Theorem 1.1 and Corollary 1.1 has been recently obtained in [8]. There, using Kubota’s formula [16, (5.72)] and induction on the dimension, the authors derive the above relations for the classical quermassintegrals of , i.e., when the relative body is the Euclidean unit ball . Here we show, by just exploiting the use of the Steiner formula (1.2), that these relations can be extended to the setting of the so-called Minkowski relative geometry, this is, when the functionals of convex bodies are evaluated in relation to an arbitrary convex body rather than . This will be shown in Section 2 of the present paper. Next, in Section 3 we prove further inequalities involving quermassintegrals of Minkowski sums of convex bodies.
2. Relations for the quermassintegrals of Minkowski sums
We start this section by showing that the sequence of (relative) quermassintegrals is convex when is a summand of . This will be proven by using the Steiner formula (1.2) jointly with the following property of the binomial coefficients of the numbers :
[TABLE]
where whenever .
Lemma 2.1**.**
Let with and . Then the sequence is convex, i.e., for every ,
[TABLE]
Equality holds if and only if .
Proof.
By the Steiner formula for the quermassintegrals (1.2) we get
[TABLE]
and thus, from (2.1), we have that
[TABLE]
Moreover, from the latter identity we may assert that if and only if . This completes the proof. ∎
At this point we would like to compare this result with Bonnesen’s inequality in the plane, which establishes that
[TABLE]
with equality if and only if for . Here is the relative inradius of with respect to , which is defined by
[TABLE]
Bonnesen proved this result for [6], being the proof of the general case due to Blaschke [4, Pages 33-36]. This inequality sharpens (in the plane) the Aleksandrov-Fenchel and the isoperimetric inequality, and there is no known generalization of it to higher dimension. Some Bonnesen style inequalities in arbitrary dimension can be found in e.g. [5, 7, 9, 10, 14, 15] (see also [16, Notes for Section 7.2] and the references therein).
Remark 2.1**.**
We notice that if are such that , which implies that , then Lemma 2.1 for (and such a pair of convex bodies) is just a simple consequence of Bonnesen’s inequality.
Now we derive a more general result for three non-necessarily consecutive (relative) quermassintegrals. For the sake of clarity, we notice that along its proof we are following the steps of [16, Pages 399, 400] (cf. (1.4)) (see also [8]). From now on, and unless we say the opposite, .
Proof of Theorem 1.1.
Let be fixed. From Lemma 2.1 it follows that
[TABLE]
We notice that the latter inequality follows from the fact that and the monotonicity and translation invariance of mixed volumes.
Thus,
[TABLE]
and hence,
[TABLE]
which yields (1.6). Moreover, (1.6) holds with equality if and only if the same holds for (2.2). Thus, the result follows from the equality case of Lemma 2.1. ∎
The following classical result provides us with a sufficient condition for a sequence of positive real numbers to be the (relative) quermassintegrals of certain convex bodies :
Proposition 2.1** ([17]).**
If the sequence of positive real numbers is log-concave, then there exist simplices such that , .
A refined version of this result can be found in [13]. The most general problem of characterizing whether a finite sequence of non-negative numbers are the mixed volumes of convex bodies remains open, except the -dimensional case for , which was solved by Heine in [12].
Using Proposition 2.1 we may assure on one hand that there exist convex bodies in for which (1.6) does not hold, and that there exist convex bodies, which are not Minkowski sums one of each other, satisfying (1.6), on the other hand. We collect the statement here, for the sake of completeness:
Proposition 2.2**.**
- i)
There are convex bodies for which (1.6) does not hold. 2. ii)
There exist pairs of convex bodies for which (1.6) does hold and such that for all .
Proof.
The idea for both assertions is to use Proposition 2.1. To prove (i) it is enough to find a log-concave sequence of positive real numbers which is not convex. We consider , whose log-concavity (cf. (1.5)) is clearly fulfilled and thus, it ensures the existence of planar simplices with and , and for which (1.6) does not hold.
To prove (ii), we argue in the same way with the sequence , which is both log-concave and convex. Since this sequence satisfies (2.2) with equality, and , the simplex cannot be a summand of the simplex . ∎
Taking Theorem 1.1 into account, together with the log-concavity of the quermassintegrals of any convex body, it is natural to ask whether the log-concavity together with the convexity (cf. Theorem 1.1) of the sequence of quermassintegrals of a Minkowski sum could provide us with better inequalities or, eventually, equalities in some known inequalities. Unfortunately, so far we have no concluding answers to this issue.
3. Further inequalities
In the following we will write, for ,
[TABLE]
to denote the (relative) Steiner polynomial of , regarded as a formal polynomial in a complex variable . Notice that, for , yields the volume of (cf. (1.1)). Moreover, we consider the derivatives of Steiner polynomials in the variable , namely
[TABLE]
for . Of particular interest for our purposes will be the quantities given by
[TABLE]
for any (and fixed ), which can be regarded as formal extensions (up to the constant ) of the value at of the -th derivative of the Steiner polynomial, since
[TABLE]
Now we prove some generalizations of the inequalities we have dealt with in the previous section. To avoid straightforward (but a bit lengthy) computations throughout the (proof of) Proposition 3.1, first we collect here some direct relations for combinatorial numbers.
Lemma 3.1**.**
Let . If then
[TABLE]
Moreover, if then
[TABLE]
Proof.
Assume first that . Then, by using recursively (2.1), we have
[TABLE]
Now, if , and using again recursively (2.1) we get that
[TABLE]
which, by the previous case, is non-negative. This finishes the proof. ∎
We note that, in terms of the quantities (see (3.1)), Lemma 2.1 yields that
[TABLE]
for every . Here we extend these inequalities to any value of , with .
Proposition 3.1**.**
Let with and . Then, for every ,
[TABLE]
Equality holds if and only if .
Proof.
From the Steiner formula for quermassintegrals (1.2), and writing , we have
[TABLE]
From Lemma 3.1 we obtain that all the summands in the above expression are non-negative. Moreover, since the sole non-zero coefficients involved are those with indexes between and , we have that if and only if . This completes the proof. ∎
Lemma 3.2**.**
Let with and . Then, for every ,
[TABLE]
Proof.
By using (2.1) we get that
[TABLE]
which, jointly with Proposition 3.1, finishes the proof. ∎
We conclude the paper by showing an extension of Theorem 1.1. Indeed, the latter is derived by setting in the following result.
Theorem 3.1**.**
Let with and . Then, for every ,
[TABLE]
Equality holds if and only if .
Proof.
The proof is completely analogous to the proof of Theorem 1.1, interchanging every involved -th quermassintegrals by , and using Lemma 3.2 instead of Lemma 2.1. Finally, (3.2) holds with equality if and only if the same holds for
[TABLE]
which is equivalent to . Then the result follows from the equality case of Proposition 3.1. ∎
Acknowledgements. We would like to thank the anonymous referee for his/her very valuable suggestions which have allowed us to considerably improve the presentation of the manuscript.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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