On the monotonicity of the isoperimetric quotient for parallel bodies

The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained.

Let K n denote the family of convex bodies in R n , i.e., of nonempty compact convex subsets of the Euclidean space R n , and let K n n be its subfamily of convex bodies with nonempty interior. Let B n be the n-dimensional unit ball and S n−1 the corresponding unit sphere. The volume of a convex body K ⊆ R n , i.e., its n-dimensional Lebesgue measure, is denoted by vol(K). The measure of its boundary, i.e., its surface area or (n − 1)-dimensional Hausdorff measure, is denoted by S(K).
For K ∈ K n n , the classical isoperimetric quotient is the ratio (1.1) I(K) = S(K) n vol(K) n−1 .
Let K, E ∈ K n . The inradius r(K; E) of K relative to E is the largest possible factor of a homothety mapping E into K, i.e., r(K; E) = sup{r ≥ 0 : there is x ∈ R n with x + r E ⊆ K}.
We prove that I(K) ≥ I(K λ ) for − r(K; E) < λ < 0 by investigating the behaviour of the isoperimetric quotient as a function of the real parameter λ. Our main result in this direction is the following (see Corollary 3.5), extending the well-known result for outer parallel bodies [12,Remark 4.4] to inner ones. Theorem 1.1. Let K ∈ K n n . Then the isoperimetric quotient I(K λ ) is monotonically decreasing for λ ∈ (− r(K; B n ), ∞).
We will also analyze the situation where the function λ → I (K λ ) is constant in some subinterval of (− r(K; B n ), ∞), characterizing the convex bodies K ∈ K n n for which this behaviour occurs. We notice that the surface area is implicitly defined with respect, or relative, to the Euclidean unit ball, see Section 2 for details. Indeed, the two magnitudes, volume and surface area, involved in the isoperimetric quotient are particular cases of the so-called quermassintegrals of a convex body, and can be defined relative to a fixed body E ∈ K n n , for which we refer the reader to Section 2 and the references therein.
In this note, for a fixed reference body E ∈ K n n -usually called gauge, we investigate the behaviour of the relative isoperimetric quotient function S(K λ ;E) n vol(K λ ) n−1 (see Section 2 for details) of the family of parallel bodies of the convex body K ∈ K n n with respect to E. In [12] (cf. [6]) the authors prove that, under certain boundary restrictions of the involved convex bodies or under certain differentiability conditions of the involved magnitudes, respectively, the relative isoperimetric quotient S(K λ ;E) n vol(K λ ) n−1 is monotonically decreasing for λ ∈ (− r(K; E), ∞). For λ ≥ 0, i.e., for outer parallel bodies, this is a direct consequence of the relative Steiner formula (2.2) [12,Remark 4.4]. We point out that in the planar case K, E ∈ K 2 2 the monotonicity in the whole domain (− r(K; E), ∞) seems to be treated as folklore, and to the best of our knowledge, there is no precise reference containing this result.
Here we prove that no assumption on the convex bodies K, E ∈ K n n is necessary to have monotonicity on the total domain (− r(K; E), ∞). Moreover, we characterize all convex bodies for which the quotient is constant in some interval of its domain (see Theorem 3.4).
In the above result, the family (K λ ) − r(K;E)≤λ≤0 of inner parallel bodies is extended by the family (K λ ) λ≥0 of outer parallel bodies. In Section 4 we introduce other natural extensions of (K λ ) − r(K;E)≤λ≤0 to parameters λ ≥ 0, based on particular Wulff shapes. Also the isoperimetric quotients of these families are monotonically decreasing (see Theorem 4.5). This allows us to characterize isoperimetrically optimal bodies K with prescribed outer normals, again relative to a gauge body E (see Section 4).
Finally, we investigate the behaviour of further quotients of magnitudes for parallel bodies intimately connected to the isoperimetric quotient, both for the classical and relative cases (see Section 5). This continues work of [12].

Background
We write K λ to denote the inner and outer parallel bodies of K ∈ K n relative to the gauge body E ∈ K n n , For K ∈ K n and u ∈ R n , h K (u) = sup x, u : x ∈ K denotes the support function of K ∈ K n , where ·, · stands for the standard Euclidean scalar product in R n (see e.g. [21,Section 1.7]). For u ∈ R n \ {0} and α ∈ R, we define the halfspace H − u,α = {x ∈ R : x, u ≤ α} [21, p. xx]. Then the above parallel bodies K λ can be equivalently defined as The so-called relative Steiner formula states that the volume of the outer parallel body K + λE is a polynomial of degree n in λ ≥ 0, The coefficients W i (K; E) are called the relative quermassintegrals of K, and they are just a special case of the more general mixed volumes, for which we refer to [21, Section 5.1] and [8, Sections 6.2, 6.3] (where [21] uses the notation V (i) (K, E) for W i (K; E)). In particular, we have W 0 (K; E) = vol(K) and W n (K; E) = vol(E). Applying (2.2) to both sides of K + (λ + µ)E = (K + λE) + µE for λ, µ ≥ 0 and equating the coefficients of µ i in both polynomials, we obtain the values of the relative i-th quermassintegrals of K + λE, namely for λ ≥ 0 and i = 0, . . . , n (cf. [8,Theorem 6.14]). If E = B n , the polynomial on the right-hand side of (2.2) becomes the classical Steiner polynomial, see [22]. Then the coefficient n 1 W 1 (K; B n ) in (2.2) happens to be the surface area S(K) if K ∈ K n n . This motivates the definition of the surface area of K relative to E ∈ K n n by (2.4) S(K; E) = nW 1 (K; E) (see e.g. [10, Section 5.1.2]) and the introduction of the relative version of (1.1). 12]). Let K, E ∈ K n n . The isoperimetric quotient of K relative to E is defined as Note that I(K; E) is invariant under homotheties of K, since the i-th relative quermassintegral is positively homogeneous of degree n − i, i.e., W i (µK; E) = µ n−i W i (K; E) for any µ ≥ 0.
Our goal is to analyze the behaviour of (2.5), focusing on the question whether it is a monotonic function in (some parts of) its domain.
A first naive approach to this matter is the natural question whether there are convex bodies K, E ∈ K n n for which the isoperimetric quotient function is constant in (− r(K; E), 0]. We observe that K − r(K;E) has no interior points, thus we need to exclude it.
Tangential bodies provide us with a positive answer to this question. 21, p. 149, text preceding Lemma 3.1.14]). Let K, E ∈ K n n be such that E ⊆ K. Then the body K is a tangential body of E if and only if through each boundary point of K there exists a supporting hyperplane of K that also supports E.
Indeed, inner parallel bodies and tangential bodies happen to be intrinsically connected by means of a homothety relation. The following result enlightens the close connection between inner parallel bodies and tangential bodies, providing us with a constant isoperimetric quotient function on the range (− r(K; E), 0) of inner parallel bodies. Theorem 2.3 ([21, Lemma 3.1.14]). Let K, E ∈ K n n be convex bodies, and let λ ∈ − r(K; E), 0 . Then K λ is homothetic to K if and only if K is homothetic to a tangential body of E.
We notice, that if K is a tangential body of E, then r(K; E) = 1. It follows that the isoperimetric quotient function I(λ) is constant for − r(K; E) = −1 < λ ≤ 0 if K is a tangential body of E.
Although the relative quermassintegral W i (·; E) : K n → R, i = 0, . . . , n − 1, is, in general, not linear under Minkowski addition and scalar multiplication, the (n − i)-th root of W i (·; E) is concave. In particular, the n-th root of the volume is a concave function. This is a consequence of the fundamental Aleksandrov-Fenchel inequality [ For i = 0, i.e., for the volume, equality holds if and only if K and L are homothetic. If E is smooth and 0 ≤ i ≤ n − 2, equality holds if and only if K and L are homothetic.
Here, a convex body is said to be smooth if it has only one supporting hyperplane at every point of its boundary. The inequality in Theorem 2.4 can be found in [21,Theorem 7.4.5], the characterization of equality for i = 0 is given in [21, Theorem 7.1.1], and the equality case for smooth E is a consequence of [21, Theorems 7.4.6 and 7.6.9].
To prove the monotonicity of the quotient function I(λ), the derivative of the volume vol(K λ ), as a function of λ ∈ (− r(K; E), ∞), turns out to be crucial. In order to deal with derivatives of the quermassintegrals W i (K λ ; E), in particular of the volume vol(K λ ) = W 0 (K λ ; E), we need the following results.
For i = 0, i.e., for the case of the volume, even more is known.

Monotonicity of the isoperimetric quotient for parallel bodies
In this section we will prove that the isoperimetric quotient function is decreasing. Our result is prepared by some auxiliary statements. The first one is an immediate consequence of formula (2.9) or Proposition 2.6.
The following statement corresponds to formula (18) of [9, Section 6] if E = B n . Although the proof is analogous to the proof in the case E = B n , we include it for the sake of completeness.
Let K, L ∈ K n n , and let K + L be homothetic to L. Then, there are α > 1 and x 0 ∈ R n such that K + L = αL + x 0 . Hence, rewriting the latter in terms of support functions Now we can prove the monotonicity of the isoperimetric quotient of the family (K λ ) λ>− r(K;E) . Theorem 3.4. Let K, E ∈ K n n be convex bodies. Then the function Moreover, the following are equivalent for all − r(K; E) < λ 0 < λ 1 < ∞: of that function exists, and is monotonically decreasing (see e.g. [19]). Hence, by (ii)⇒(iii). Let K λ0 and K λ1 be homothetic. Then, by Lemma 3.2, we can ensure that the inner parallel body K λ0 = K λ1 ∼ |λ 1 − λ 0 |E of K λ1 is homothetic to K λ1 . Now Theorem 2.3 shows that K λ1 is homothetic to a tangential body of E.
(iv)⇒(i) is trivial. Now we deal with the last claim. For that, let λ 1 > 0, and assume that any of the equivalent assertions (i)-(iv) holds. By (iv), I(0) = I(λ 1 ), since − r(K; E) < 0 < λ 1 . Then, by (ii), K 0 = K is homothetic to K λ1 = K + λ 1 E, which together with Lemma 3.3 yields that K is homothetic to λ 1 E and thus K is homothetic to E itself.
As a corollary, we obtain that the classical isoperimetric quotient is monotonically decreasing, by setting E = B n .
Corollary 3.5. For every K ∈ K n n and E = B n , the function is monotonically decreasing on (− r(K; B n ), ∞).

Monotonicity for related families of bodies and isoperimetric problems
In this section we introduce some new (one-parameter) families of bodies related to the family (K λ ) λ≥− r(K;E) of parallel bodies. First we recall that a body K is determined by a set Ω ⊆ S n−1 if K =  The elements of U(K) are also known in the literature as extreme normal vectors of K. The set U(K) need not be closed; e.g. when K ∈ K 2 2 is a semicircle. First we prove that in the definition (2.1) of inner parallel bodies of K ∈ K n n relative to E ∈ K n n , we can replace the complete sphere S n−1 in the intersection by any subset Ω that determines K. Lemma 4.2. Let K, E ∈ K n n and let Ω ⊆ S n−1 determine K. Then (4.1) Proof. Let K, E ∈ K n n and let Ω ⊆ S n−1 determine K. By (2.1), it is evident that K λ ⊆ u∈Ω H − u,hK (u)+λhE (u) . For the verification of the reverse inclusion, let x ∈ u∈Ω H − u,hK (u)+λhE (u) be arbitrary. Then, x ∈ H − u,hK (u)−|λ|hE (u) for all u ∈ Ω, i.e., x, u ≤ h K (u) − |λ|h E (u) for all u ∈ Ω. We observe that the latter yields that x + |λ|E ⊆ H − u,hK (u) for every u ∈ Ω, since clearly, for any e ∈ E and u ∈ Ω, x + λe, u ≤ h K (u). Consequently, where the last identity follows from the fact that Ω determines K. Hence, x ∈ K λ . Remark 4.3. Given a set Ω ⊆ S n−1 that contains the origin in the interior of its convex hull, we define the set Observing that E ⊆ E Ω and that h E (u) = h E Ω (u) for u ∈ Ω, it is clear that E Ω is a tangential body of E. Indeed, it follows from the definition that E Ω is the smallest tangential body of E that is determined by Ω.
As a particular case of the latter we can obtain the so-called form body of K relative to E, for any K ∈ K n n . This is defined (cf. [21, p. 386]) as We notice that U (K * ) ⊆ cl(U (K)) and that the inclusion may be strict [20, Lemma 4.6 and the preceding example]. For fixed K, E ∈ K n n and Ω ⊆ S n−1 determining K, Lemma 4.2 motivates the introduction of the following one-parameter family of convex bodies associated to K, Here the representation (4.1) is naturally extended to λ > 0. The sets K(Ω, λ), λ ∈ [− r(K; E), ∞), are, according to [21, Section 7.5], Wulff shapes or Alexandrov bodies associated with the pairs (Ω, f λ ) where f λ : Ω → R, u → h K (u) + λh E (u). Our aim is to prove that the relative isoperimetric quotient functions defined for these new oneparameter families of bodies are also decreasing. Before turning to that, we prove the following lemma, which will be useful for later considerations.
Proof. Let K, E ∈ K n n , and let Ω ⊆ S n−1 so that U (K) ⊆ cl(Ω). (i) follows directly from (2.1). (ii): The first equality is the definition of K λ . The second one follows from Lemma 4.2 and the observation that h E (u) = h E Ω (u) for all u ∈ Ω. Putting Ω = U(K) as a particular case, we obtain the last equality.
Next we state the result about the relative isoperimetric quotient for the family K(Ω, λ) which we aim to prove.  Proof. Let K, E ∈ K n n and let Ω ⊆ S n−1 determine K. From the definition of K(Ω, Λ) it follows that K(Ω, Λ) is determined by Ω. It follows also from the definition, that h K(Ω,Λ) (u) ≤ h K (u) + Λh E (u) for all u ∈ Ω. Using the latter, together with To prove the reverse inclusion, let x ∈ K(Ω, λ). Then x ∈ H − u,hK (u)+λhE (u) for every u ∈ Ω. As in the proof of Lemma 4.2, it follows that for every u ∈ Ω. Since the last inclusion holds for all u ∈ Ω, we get x + |λ − Λ|E ⊆ K(Ω, Λ). That is, x ∈ (K(Ω, λ)) λ−Λ .
On the other hand, we have From the latter it follows that h K(Ω,λ) (u) ≥ h K (u) + λh E (u) for any u ∈ S n−1 . Thus, Lemma 4.8. Let K, E ∈ K n n , let Ω ⊆ S n−1 determine K, and let λ > 0. If K(Ω, λ) is homothetic to K then K is homothetic to E Ω .

Now we can prove Theorem 4.5
Proof of Theorem 4.5. Let K, E ∈ K n n and let Ω ⊆ S n−1 determine K. In order to prove that the relative isoperimetric quotient function I Ω (λ) = S(K(Ω,λ);E) n vol(K(Ω,λ)) n−1 is decreasing, we will interpret the assertion in terms of the relative (to E) isoperimetric function I(λ) = I(L λ ; E) = S(L λ ;E) n vol(L λ ) n−1 for an appropriate convex body L, since for this function we have already proven in Theorem 3.4, that it is decreasing.
Since this can be done for all Λ > − r(K; E), the proof of the monotonicity and the equivalence of (i)-(iv) is complete. It only remains to prove the claim concerning λ 1 > 0.
Next we consider the asymptotic behavior of K λ and K(Ω, λ) for λ → ∞. We obtain in particular that the inclusions in (4.4) may be strict for large λ, since the respective limit shapes can be different. We remark that the convergence result and the limit value for K λ can, in its essence, be found in [20, p. 56].
Proof. The convergences of the bodies follow from their definitions, i.e., from (2.1) and (4.2). The remaining claims I(E; E) = n n vol(E) and I E Ω ; E = n n vol E Ω have been shown in the proofs of Theorems 3.4 and 4.5, respectively. Remark 4.12. Suppose that Ω determines both K and E. Then the families (K λ ) λ≥− r(K;E) and (K(Ω, λ)) λ≥− r(K;E) agree for λ ≤ 0 and have the same limit shape E = E Ω . Nevertheless, they do not necessarily coincide. In fact, [20, pp. 23-24] and [12,Section 3] give examples of polytopes K, E ∈ K 3 3 such that U(K) = U(E) U(K + E). Then K 1 = K(U(K), 1), because K(U(K), 1) is determined by U(K) whereas K 1 = K + E is not.
We come to isoperimetrically optimal bodies. Corollary 4.14. Let E ∈ K n n and let Ω ⊆ S n−1 be a set that contains the origin in the interior of its convex hull.
Then a convex bodyK ∈ K n n is a minimizer of the relative isoperimetric quotient I(K; E) among all convex bodies K ∈ K n n that are determined by Ω if and only ifK is homothetic to the tangential body E Ω of E. In particular, that minimal quotient is I E Ω ; E = n n vol E Ω .
Proof. Let K ∈ K n n be determined by Ω. By  For that case Corollary 4.14 is a well-known result [21, p. 385], that goes back to Lindelöf and Minkowski [15,17] for finite Ω and to Aleksandrov [1] for general Ω.
For E = B n and Ω = S n−1 , we obtain the isoperimetric inequality for arbitrary convex bodies.

Isoperimetric-type quotients of quermassintegrals
This section is motivated by the following monotonicity result from [12]. . Let K, E ∈ K n n , let 0 ≤ i < j < n, and suppose that K belongs to the class R j . Then the function is monotonically decreasing on (− r(K; E), ∞).
Here the classes R j are defined by a differentiability condition of the functions λ → W i (K λ ; E) on [− r(K; E), ∞). 11]). Let E ∈ K n n and let p be an integer, 0 ≤ p ≤ n − 1. A convex body K ∈ K n belongs to the class R p if, for all 0 ≤ i ≤ p and for − r(K; E) ≤ λ < ∞, the following equalities hold where the first equation is to be dropped when λ = − r(K; E).
We remark that the above definition is natural taking (2.7) into account. Proposition 2.6 yields R 0 = K n for any E ∈ K n n . Clearly, and the inclusions are strict in general, as was shown in [11]. The case i = 0, j = 1 in Proposition 5.1 gives the monotonicity of I(λ) proven in Theorem 3.4, but only under the additional assumption K ∈ R 1 . In a similar way as we have relaxed the assumptions (to none) in the case i = 0, j = 1 in Theorem 3.4, in the next we shall see that, more generally, the assumption K ∈ R j from Proposition 5.1 can be relaxed by K ∈ R j−1 . We need the following lemma. Lemma 5.3. Let 0 ≤ i ≤ n − 1, E ∈ K n n , K ∈ R i and − r(K; E) < λ < ∞. Then Proof. The result follows immediately from the definition of the class R i and the standard rules of differentiation.
Proposition 5.4. Let 0 ≤ i ≤ n − 2, K, E ∈ K n n and K ∈ R i . Then the function is monotonically decreasing on (− r(K; E), ∞). Moreover, if E is smooth, the following are equivalent for all − r(K; E) < λ 0 < λ 1 < ∞: . If E is smooth and λ 1 > 0, conditions (i)-(iv) are satisfied if and only if K is homothetic to E and, consequently, if and only if I i (λ) = vol(E) for all λ ∈ (− r(K; E), ∞).
The case i = n − 1 is excluded, since I n−1 (λ) = W n (K λ ; E) = vol(E) is constant. Before dealing with the proof, we introduce a refined definition of tangential bodies (of a fixed convex body), for which we need the notion of p-extreme supporting hyperplanes, 0 ≤ p ≤ n − 1. Given a convex body K ∈ K n n , a supporting hyperplane H u,hK (u) of K is called (n − p − 1)-extreme [21, p. 85] if its outer normal vector u ∈ R n \ {0} cannot be represented as a sum of n − p + 1 linearly independent outer normal vectors at the same boundary point of K.
Definition 5.5 ([21, p. 86]). Let K, E ∈ K n n be such that E ⊆ K, and let p ∈ {0, . . . , n − 1}. Then K is a p-tangential body of E if each (n − p − 1)-extreme supporting hyperplane of K is a supporting hyperplane of E.
We observe that K is a tangential body of E in the sense of Definition 2.2 if and only if K is just an (n − 1)-tangential body of E [21, pp. 86, 149].
Remark 5.6. Let E ∈ K n n . (i) Every p-tangential body of E is also a q-tangential body of E whenever 0 ≤ p < q ≤ n−1.
(ii) The only 0-tangential body of E is E itself.
(iii) A tangential body K of E belongs to the class R p if and only if K is an (n − p − 1)tangential body of E [11, Theorem 1.3].
Proof of Proposition 5.4. Let 0 ≤ i ≤ n − 2, and let K ∈ K n n lie in the class R i . By Lemma 5.3 and the assumption of K ∈ R i , the derivative d dλ W i (λ) We notice that the statements (i)-(iv) are the exact analogues of the statements (i)-(iv) in Theorem 3.4, with the only addendum of K being an (n − i − 1)-tangential body of E instead of just a tangential body of E. However, these two assertions are equivalent for bodies K ∈ R i by Remark 5.6. Hence the proof of the equivalences of (i)-(iv) can follow the respective steps of the proof of Theorem 3.4.
For the last part of the proof, we can proceed in analogy to the proof of Theorem 3.4, too. Indeed, note that from the homogeneity of quermassintegrals, once obtained K λ = (1 + λ)E, it follows that I i (λ) = vol(E) by using that W j (E; E) = vol(E) for any 0 ≤ j ≤ n.
We observe that for i = 0, we recover Theorem 3.4 (notice the multiplicative constant n from (2.4)), since the class R 0 consists of all convex bodies. Now we obtain the announced improvement of [12, Proposition 4.3] (i.e., of Proposition 5.1) as a corollary.
If E is smooth and λ 1 > 0, conditions (i)-(iv) are satisfied if and only if K is homothetic to E and, consequently, if and only if I i,j (λ) = vol(E) j−i for all λ ∈ (− r(K; E), ∞).
The following result is now obtained in the same way as Corollary 4.14 was proven using Theorem 4.5.
Corollary 5.8. Let 0 ≤ i < j < n and let E ∈ K n n be smooth. Then a convex bodyK ∈ R j−1 ∩ K n n is a minimizer of the quotient Wj (K;E) n−i Wi(K;E) n−j among all convex bodies K ∈ R j−1 ∩ K n n if and only if K is homothetic to E.
The assumption K ∈ R j−1 in Proposition 5.7 and Corollary 5.8, respectively, is essential in our proof. However, so far we do not have an example of a body K ∈ K n n \R j−1 that does not satisfy the claims of Proposition 5.7 or Corollary 5.8.
Remark 5.9. Theorem 3.4 on the family (K λ ) λ>− r(K;E) of parallel bodies gave rise to the analogous Theorem 4.5 on the families (K(Ω, λ)) λ>− r(K;E) , since the last families could be interpreted as inner parallel bodies by Lemma 4.6. In a similar way, results of the present section imply analogues concerning the families (K(Ω, λ)) λ>− r(K;E) . Then conditions K ∈ R i have to be replaced by K(Ω, Λ) ∈ R i for all Λ > − r(K; E).