Sausage body is a solution for a reverse isoperimetric problem

# A sausage body is a unique solution for a reverse isoperimetric problem

Roman Chernov Kostiantyn Drach  and  Kateryna Tatarko Jacobs University Bremen, Research I, Campus Ring 1, 28759 Bremen, Germany University of Alberta, 677 Central Academic Building, Edmonton, AB, T6G 2H1 Canada
###### Abstract.

We consider the class of -concave bodies in ; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius (a sausage body) is a unique volume minimizer among all -concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique minimizer is a ball. We solve the reverse isoperimetric problem by proving a reverse Bonnesen-style inequality, the second main result of this paper.

###### Key words and phrases:
-concavity; -convexity; reverse isoperimetric inequality; quermassintegral; -concave polytope
###### 2000 Mathematics Subject Classification:
52A30, 52A38; 53A07, 52A39, 52A40, 52B60
The research of the first two authors was partially supported by the advanced grant “HOLOGRAM” of the European Research Council (ERC), which is gratefully acknowledged. The second author is also grateful to Cornell University for their hospitality during his visit there. We thank Alexander Litvak and Vlad Yaskin for reading some parts of the manuscript and providing some useful remarks.

## 1. Introduction

The classical isoperimetric inequality states that if is an arbitrary domain in with volume and surface area , then

 (1.1) Voln+1(K)⩽Voln(∂K)n+1n(ωn+1)1n(n+1)n+1n,

where is the volume of the unit ball in . It is known that equality in (1.1) holds if and only if is a ball. In other words, the classical isoperimetric inequality asserts that among all domains with a given surface area, the ball has the smallest possible volume.

Inequality (1.1) has a long and beautiful history, and has been generalized to a variety of different settings (see, for example, surveys [BZ, Ro]). The distinctive point of almost all of these generalizations is that the extreme object is always a ball, as the most symmetric body. On the other hand, the problem can be looked at from a different point of view: under which conditions can one minimize the volume among all domains of a given constraint (such as of a given surface area, etc.)? Questions of such type are known as reverse isoperimetric problems, and have been actively studied recently.

The naive attempt of minimizing volume among all sets of a given surface area will clearly lead to a trivial result: the -dimensional volume is zero for every set with empty interior. Therefore, we must consider a family of sets with additional conditions imposed in order to obtain a well-posed reverse isoperimetric problem. One of the natural conditions is convexity or strict convexity.

One of the first results on the reverse isoperimetric problem is due to Keith Ball. In his celebrated works [Bal1, Bal2] he showed that among all convex bodies in (modulo affine transformations), the standard simplex has the smallest volume for a given surface area; if the bodies are additionally assumed to be symmetric, then the cube is an extreme object. The equality case in Ball’s reverse isoperimetric inequalities was completely settled later by Barthe [Ba]. Observe that for Ball’s approach the minimizers are no longer balls.

Another approach towards obtaining a reverse isoperimetric inequality was recently taken in [PZh], where the authors provided a lower bound on the area enclosed by a convex curve in terms of its length and the area of the domain enclosed by the locus of curvature centers of . The authors also showed that equality is attained only for a disk. In this respect, the results in [PZh] do not follow the philosophy of a reverse isoperimetric problem. See also [XXZZ], but again these results, although called ‘reverse’, do not follow the philosophy of a reverse isoperimetric problem.

A different approach towards reversing the classical isoperimetric inequality is by assuming some curvature constraints for the boundary. It was pioneered by Howard and Treibergs [HTr] who proved a sharp reverse isoperimetric inequality on the Euclidean plane for closed embedded curves whose curvature , in a weak (or viscosity) sense, satisfies , and whose lengths are in . In [Ga], Gard extended this result to surfaces of revolution that lie in and whose principal curvatures, again in a weak sense, are bounded in absolute value by , and the surface areas are not too big. Note that the mentioned curvature restrictions do not imply convexity.

At the same time, motivated by the study of strictly convex hypersurfaces in Riemannian spaces (see, for instance, [Bor1, BDr1]), Borisenko and Drach in a series of papers [BDr2, BDr3, Dr1] obtained two-dimensional reverse isoperimetric inequalities for so-called -convex curves, i.e. curves whose curvature , in a weak sense, satisfies . Recently, these results were generalized in [Bor2] for -convex curves in Alexandrov metric spaces of curvature bounded below. The result of Borisenko completely settles the reverse isoperimetric problem for -convex curves.

-convexity is a notion that can be easily transferred to higher dimensions. In particular, a convex body in is -convex if the principal curvatures of the boundary of the body are uniformly bounded, in a weak sense, by , i.e.  for all (we refer to [BM, BGR, Dr2] for various results concerning the geometry of multidimensional -convex bodies). It is worth pointing out that the reverse isoperimetric problem for -convex bodies has a non-trivial solution in any dimension, although for dimensions greater than two it is a hard problem that is still widely open.

In this paper we consider a notion, in a sense dual to the notion of -convexity. In particular, we consider so-called -concave bodies in . These are the convex sets such that the principal curvatures of their boundaries satisfies for all (in a viscosity sense, see Definition 1.2). For -concave bodies we completely solve the reverse isoperimetric problem in any dimension. This is the first result on the reverse isoperimetric problem in , besides the celebrated results of Ball and their various extensions, where the inequality is not restricted to curves or surfaces. Moreover, our methods allow us to prove the full family of sharp inequalities involving quermassintegrals of a convex body.

### 1.1. Motivation.

Part of our motivation, besides previously mentioned work on the reverse isoperimetric problem for -convex domains due to Borisenko and Drach, came from results on the so-called Will’s conjecture.

If is a planar convex body with inradius , then the inequality

 Vol2(K)⩽rVol1(K)−r2π

is called Bonnesen’s inradius inequality. Equality holds for the sausage body, that is, the Minkowski sum of a line segment and a circle with radius . An extension of Bonnesen’s inradius inequality to higher dimensions was conjectured by Wills [Wi] in 1970. He conjectured that

 Voln+1(K)⩽rVoln(∂K)−nrn+1ωn+1,

for every convex body with inradius . This conjecture was proven independently by Bokowski [Bo] and Diskant [Di]. Although the same inequality with the circumradius of substituting is not true in dimensions greater than two (see [Di, He]), Bokowski and Heil [BH] showed that for higher dimensions, in fact, the inequality sign is reversed:

 (1.2) Voln+1(K)⩾2RnVoln(∂K)−(n+2)Rn+1nωn+1.

In [BH] inequality (1.2) was obtained as a corollary of the following more general result

###### Theorem 1.1 ([Bh]).

For an arbitrary convex body with circumradius , the inequalities

 (1.3) cijkRiWi(K)+cjkiRjWj(K)+ckijRkWk(K)⩾0,

hold for every , where .∎

Here is the quermassintegral of order of the convex body , (see the next subsection and Section 2 for precise definitions). Quermassintegrals can be viewed as geometric quantities assigned to a convex body that are a higher-dimensional generalization of the integral curvature of a closed curve, and can be explicitly calculated in terms of the principal curvatures of , provided is sufficiently smooth (see (2.1)). The quermassintegrals of different order provide a natural embedding of the volume , the surface area and the volume of the unit ball into the family for which (up to the constant) these are respectively, the zeroth, the first, and the -th element. Therefore, (1.2) is a special case of (1.3) with , and .

The form of the Bokowski–Heil inequality (1.2) inspired the statement of our main result, Theorem A, although we use different techniques for the proof. It appears that, having a natural inclusion of the volume, the surface area and the volume of the unit ball into the family of quermassintegrals helps to solve the reverse isoperimetric problem for -concave bodies in for every .

### 1.2. The main results.

Recall that a convex body in the Euclidean space is a compact convex set with a non-empty interior. In this paper balls will be closed sets.

###### Definition 1.2 (λ-concave body).

For a given , a convex body is called -concave if for every there exists a ball (called a supporting ball at ) of radius passing through in such a way that

 (1.4) B1/λ,p∩U(p)⊆K∩U(p)

for some small open neighborhood of .

Note that since is assumed to be convex, if is -concave then a supporting ball is unique at every point. As for the nomenclature, compare it to the notion of -convexity (see [BGR, BDr1, Dr2]), for which inclusion (1.4) is reversed (see also the discussion in Subsection 4.2).

If the boundary of a convex body is at least -smooth, then is -concave if and only if the principal curvatures for all are non-negative and uniformly bounded above by , i.e.  for every and . Equivalently, in the smooth setting -concavity can be expressed in terms of uniformly bounded normal curvature. Let be a point, be a vector, be the outward pointing normal to at , and be the two-dimensional plane through spanned by and . The normal curvature of at the point in the direction of is defined as

 kn(p,v):=κ(p),

where is the curvature of the planar curve at the point . Using this notion, a convex body with smooth boundary is -concave if and only if uniformly over and . In general, is -concave if the uniform bound on the normal curvatures is satisfied in the viscosity sense (see [BGR, Definition 2.3] for the similar approach).

Recall that for a convex body the quermassintegral of order (denoted by with ) arises as a coefficient in the polynomial expansion

 Voln+1(K+tB)=n+1∑i=0(n+1i)Wi(K)ti

known as the Steiner formula; here is the unit Euclidean ball in and ‘’ stands for the Minkowski addition; see Section 2 for details.

###### Definition 1.3 (λ-sausage body).

A -sausage body in is the convex hull of two balls of radius (see Figure 1).

We are now ready to state the main results of the paper.

###### Theorem A (Reverse quermassintegrals inequality for λ-concave bodies).

Let be a convex body. If is -concave, then

 (1.5) (k−j)Wi(K)λi+(i−k)Wj(K)λj+(j−i)Wk(K)λk⩾0

for every triple with . Moreover, equality in (1.5) holds if and only if is a -sausage body.

Since , and , inequality (1.5) for , and immediately implies the following result:

###### Theorem B (Reverse isoperimetric inequality for λ-concave bodies).

Let be a convex body. If is -concave (for some ), then

 (1.6) Voln+1(K)⩾Voln(∂K)nλ−ωn+1nλn+1,

where is the volume of the unit ball in . Moreover, equality holds if and only if is a -sausage body.∎

Theorem A (and hence Theorem B) for and was first proven using different techniques in the bachelor thesis of the first author [Ch]. It should be pointed out that Theorem B for was announced earlier in [BDr3], although the authors did not provide a proof.

## 2. General background on quermassintegrals and convex geometry

In this section we present some background material and auxiliary lemmas towards the proof of the main result.

The Minkowski addition of two convex bodies and in is defined by

 K+L:={x+y:x∈K,y∈L}.

One can rewrite the definition in the following form

 K+L=⋃y∈L(K+y);

that is can be viewed as the set that is covered if undergoes translations by all vectors in . Since and are convex, then is also convex. For a parameter , the Minkowski sum , where is the unit ball in , is called the outer parallel body for . The Minkowski difference of convex bodies and is defined by

 K−L:={x∈Rn+1:L+x⊂K}.

Similarly to the operation of addition, we can rewrite the definition of Minkowski difference in the form

 K−L=⋂y∈L(K−y).

For a parameter , the Minkowski difference is called the inner parallel body.

For a convex body , the -dimensional volume of its outer parallel body is a polynomial in , and by the classical Steiner formula,

 Voln+1(K+tB)=n+1∑i=0(n+1i)Wi(K)ti,

where is the quermassintegral of order of the convex body . If the boundary of the body is at least -smooth, and hence the principal curvatures are well-defined almost everywhere on , then

 (2.1) Wj(K)=1(n+1)(nj−1)∫∂Kσj−1dx for 1⩽j⩽n+1,

where and

 σl=∑1⩽i1<⋯

is the -th symmetric function of principal curvatures. By convention, is equal to the -dimensional volume of the body. It follows from (2.1) that is the -dimensional volume (surface area) of , and is the -dimensional volume (surface area) of the unit sphere . Recall that , where is the volume of a unit -ball in ; hence .

We will need the following generalization of the Steiner formula for inner parallel bodies (see [Sch, p. 225]):

 Wq(K−B)=n+1−q∑i=0(−1)i(n+1−qi)Wq+i(K)

for every . In particular, for we have

 (2.2) Voln+1(K−B)=n+1∑i=0(−1)i(n+1i)Wi(K).

The following continuity result [G, p. 399] will be useful later on for the proof of Theorem A.

###### Proposition 2.1 (Continuity of quermassintegrals).

Let be a sequence of convex bodies. Suppose

 limi→∞Ki=K

in the Hausdorff metric. Then

 limi→∞Wj(Ki)=Wj(K)

for every .∎

###### Definition 2.2 (Core of a λ-concave body).

A core of a -concave body is the set , where “” denotes the Minkowski difference of convex sets, is a ball of radius .

(Compare this notion to the notion of a kernel of a convex body [SY, p. 374].)

It is easy to see that is a convex set in ; however, the core is not necessarily -concave, even more, is not necessarily a convex body in . Recall that the affine hull of a convex set is the affine subspace of least dimension that contains . We will call the dimension of the core (denoted by ) to be the dimension of the affine hull of . Clearly, is a convex body if its dimension is .

The classical result due to Blaschke implies that local condition (1.4) is in fact global (see [BS, Mi], and [Bla] for the original result of Blaschke).

###### Theorem 2.3 (Blaschke’s ball rolling theorem of λ-concave bodies).

Let be a -concave body. Then

 B1/λ,p⊆K

for every point and every supporting ball at .∎

Using Blaschke’s ball rolling theorem it is easy to prove the following approximation result. We will say that a -concave body is a -concave polytope if is a polytope, possibly of lower dimension. Note that a -concave polytope is not a polytope, but rather an outer parallel body (at distance ) for a polytope.

###### Proposition 2.4 (Approximation of λ-concave bodies).

Let be a -concave body. Then there exists a sequence of -concave polytopes such that , for all , and

 (2.3) limi→∞Pi=K

in the Hausdorff metric.

###### Proof.

It is known that a convex body can be approximated from inside in the Hausdorff metric by convex polytopes with vertices on [Sch, p. 67]. Let be such a sequence. Since tends to as , we can assume that , after possibly passing to a subsequence. Moreover, we can further refine this sequence into a sequence which is nested, that is (for example, by choosing to be the convex hull of and , to be the convex hull of , and , and so on).

Let be the vertex set of . By Blaschke’s ball rolling theorem (Theorem 2.3), the union of all supporting balls of radius at points in lies in :

 (2.4) Vi:=⋃p∈V′′iB1/λ,p⊆K.

Define to be the convex hull of . Clearly, is a -concave polytope. Moreover, by construction, , , and ; the later inclusion follows from (2.4). Moreover, since supporting balls are uniquely defined by the corresponding points of support, it follows that for all . Finally, since and , we obtain (2.3). ∎

## 3. Proof of the reverse quermassintegral inequality for λ-concave bodies (Theorem A)

In order to prove Theorem A, we will use an approximation of a given -concave body with -concave polytopes. Therefore, we start by establishing the following proposition.

###### Proposition 3.1 (Reverse quermassintegral inequality for λ-concave polytopes).

Theorem A holds for -concave polytopes.

###### Proof.

Let be a -concave polytope. We want to show that inequality (1.5) for is strict unless is a -sausage body. Since the left-hand side of (1.5) is scale-invariant, without loss of generality we can assume .

Consider the core of ; define

 (3.1) m:={dimPc−1, if % dimPc=n+1,dimPc, if dimPc⩽n.

By construction, , where is a unit ball in . The latter equality provides a complete description of the boundary of in terms of principal curvatures, and hence simplifies computation. In particular, is decomposed as

 (3.2) ∂P=Cl(m⨆s=0Fn−s)

into open sets such that on exactly principal curvatures equal to , all the rest, namely , equal to (here stands for the closure of a set in ). Each is an open subset of the outer parallel set of the union of -dimensional facets of , and hence each connected component of is a measurable subset of the cylinder of non-zero -dimensional Lebesgue measure. Note that the intersection of any two connected components of and has -dimensional Lebesgue measure zero.

Plugging (3.2) into (2.1), we get

 (3.3) Wi(P)=1(n+1)(ni−1)∫∂Pσi−1dx=1(n+1)(ni−1)m∑s=0∫Fn−sσi−1dx

for every .

For define . Observe that it might happen that some of vanish. For example, for a -sausage body its core is just a segment, and hence for all . Since we know the values of principal curvatures at every point on , we obtain

 (3.4) ∫Fn−sσi−1dx=(n−si−1)as,

where, by convention, if . Since is convex, its Gauss image is equal to , hence (recall that is the volume of the -dimensional unit ball in ). Plugging (3.4) into (3.3), we obtain

 (3.5) Wi(P)=ωn+1+1(n+1)(ni−1)m∑s=1(n−s)i−1as

for every integer between and .

In order to prove (1.5), we partly follow ideas in [BH, Theorem 2]. We claim that it suffices to show that (1.5) is true for any three consecutive indices, i.e. to show that

 Ei(P):=Wi−1(P)−2Wi(P)+Wi+1(P)

is non-negative for every . Indeed, if (1.5) is true for every triple , then for a triple applying it repeatedly, we get

 Wk−Wk−1⩾Wk−1−Wk−2⩾⋯⩾Wj−Wj−1⩾⋯⩾Wi+1−Wi.

Estimating the sum of the first and the last differences in this sequence, we get

 (Wk−Wk−1)+…+(Wj+1−Wj) ⩾(k−j)(Wj−Wj−1), (Wj−Wj−1)+…+(Wi+1−Wi) ⩽(j−i)(Wj−Wj−1)

(here we simplify our notation by setting ). Performing cancellation and dividing both inequalities by and respectively, we obtain

 Wk−Wjk−j⩾Wj−Wj−1⩾Wj−Wij−i,

and hence for every ordered triple with . This is equivalent to (1.5), and the claim follows.

Let us now turn to showing that

 (3.6) Ei(P)⩾0 for every i∈{1,…,n}.

We consider two cases: and ; this is done because (3.3) does not cover the latter case, and Steiner formula (2.2) should be used instead.

Case 1: . Using (3.5) and performing cancellation along the way, we obtain

 Ei(P)=1n+1m∑s=1as⋅bs,i, where bs,i:=(n−si−2)(ni−2)−2(n−si−1)(ni−1)+(n−si)(ni).

In order to establish (3.6), it suffices to show that

 (3.7) bs,i⩾0 for all i∈{1,…,n} and s∈{1,…,m}

(recall that ). There are three possibilities: either , and hence all three terms in are non-zero, or , and hence the last term in is zero, or, finally, . For the last possibility it is clear that . In the second to last possibility we have

 bs,i=bs,n−s+1=(n−sn−s−1)(nn−s−1)−2(n−sn−s)(nn−s)=s−1(ns),

which is non-negative for every . Therefore, (3.7) holds true for , and hence we are left with the first possibility . In this case we compute:

 bs,i =(n−s)!n!((n−i+2)!(n−i+2−s)!−2(n−i+1)!(n−i+1−s)!+(n−i)!(n−i−s)!) =(n−s)!(n−i)!n!(n−i+2−s)!((n−i+2)(n−i+1)−2(n−i+1)(n−i+2−s) +(n−i+1−s)(n−i+2−s))=(n−s)!(n−i)!n!(n−i+2−s)!s(s−1).

Therefore, for (assuming , that is when the first possibility occurs). Claim (3.7) follows; this claim implies (3.6) for , as desired. Note that unless . We will use this observation below to treat the equality case.

Case 2: . Since , by definition, is the volume of , we will use Steiner formula (2.2) to estimate . Because , from (2.2) we obtain

 (3.8) W0(P)+n+1∑j=1(−1)j(n+1j)Wj(P)⩾0

Plugging (3.5) into (3.8) we get

 0 ⩽W0(P)+ωn+1⋅n+1∑j=1(−1)j(n+1j)+n+1∑j=1(−1)jjm∑s=1(n−sj−1)as =W0(P)−ωn+1+m∑s=1(n+1∑j=1(−1)jj(n−sj−1))as,

where in the last step we used the identity Hence, again using (3.5),

 E1(P)=W0(P)−2W1(P)+W2(P)⩾m∑s=1as⋅cs,

where

 cs :=−n+1∑j=1(−1)jj(n−sj−1)−2n+1+n−sn(n+1) =−n+1∑j=1(−1)jj(n−sj−1)−n+sn(n+1).

Similar to (3.7), in order to prove (3.6) for it suffices to show that

 (3.9) cs⩾0 for all s∈{1,…,m}.

Observe that, in fact,

 n+1∑j=1(−1)jj(n−sj−1)=n−s+1∑j=1(−1)jj(n−sj−1)

because the remaining terms vanish. Moreover,

 n−s+1∑j=1(−1)jj(n−sj−1)=1n−s+1n−s+1∑j=1(−1)j(n−s+1j)=−1n−s+1.

Therefore, putting all together we obtain

 cs=1n−s+1−n+sn(n+1)=s(s−1)n(n+1)(n−s+1),

which implies (3.9) for all (recall that ). Claim (3.9) yields (3.6) for . This finishes the proof of Case 2.

The conclusions of Case 1 and Case 2 together imply claim (3.6). In turn, as we showed above, (3.6) implies inequality (1.5) for an arbitrary ordered triple , as desired. The inequality part of Proposition 3.1 is proven.

Let us now turn to showing that equality in (1.5) is possible if and only if is a -sausage body. The ‘if’ part is trivial, thus we focus on the ‘only if’ part.

Equality case. Suppose we have equality in (1.5) for some ordered triple . From our reduction to three consecutive indices in the proof of inequality for it follows that we must have equality for the triple , that is we must have .

If , then we must have equality throughout our computation in Case 1. In particular, we must have for the given and all . But as we showed (see the remark at the end of Case 1), for . Therefore, we must have for all , which immediately implies that does not contain open regions where at least two principal curvatures are zero. This is only possible if is a segment (possibly degenerate to a point), and hence is a -sausage body.

If , then similarly we must have for every , and moreover, we must have equality in (3.8). The latter is possible only if the -dimensional volume of the core of is zero (hence is at most ). Again, as we showed, for we have , thus must necessarily vanish for . The same conclusion as in the previous paragraph follows.

Summing up the cases and we obtain that for inequality (1.5) turns into equality if and only if is a -sausage body. This finishes the proof of the equality part of Proposition 3.1, and together with the proven inequality part concludes the proof. ∎

The proof of Theorem A, similarly to the proof of Proposition 3.1, naturally splits into two parts: proving the inequality (the inequality part), and proving that equality is attained only for -sausage bodies (the equality part). As we already mentioned, we will establish the inequality part by using an approximation of a -concave body by -concave polytopes (Proposition 2.4), and then applying the continuity result of Proposition 2.1. This is a straightforward part of the proof. Establishing the equality case, on the other hand, involves a more delicate argument. For it we will need the following technical lemma, which gives us control over the -volume (area) of ‘the least curved’ domains on the boundary of a given -concave polytope. Recall that if is a -concave polytope, then the boundary admits decomposition (3.2) into open sets comprised of points at which principal curvatures equal to and principal curvatures equal to .

###### Lemma 3.2 (Volume dependence for facets of highest dimension).

Let be a -convex polytope, and let . Then

where is the union of all -dimensional facets of .

###### Remark.

In Lemma 3.2 we assume that is a pair of points at distance two and is the counting measure; hence .

###### Remark.

It is useful to connect and : if , then and ; however, if , then , while .

###### Proof.

Let be a point, and be the set of all supporting hyperplanes to at . Since is convex, we have if and only if and lie in the interior of a common face of (in the topology of the affine hull of this face). Therefore, if is the interior of an -dimensional facet of , then we can write , , and this set is well-defined. The set comprises of all supporting (to ) hyperplanes that contain . Write for the set of unit outward pointing normal vectors for the hyperplanes in ; the set can be identified with a closed subset of (for example, by introducing the coordinate system such that lies in the plane and lies in the set ; in this coordinate system the coordinates of the vectors in will have first entries equal to zero; the set is closed because each vector in gives a point in , taking limits preserves this property). Finally, put with being the interior of a set in the topology of ; is an open subset of . Since is convex, it follows that if and only if . From the definition of the Minkowski addition of and the unit ball (recall that ) it follows that

 Fn−d=⨆gf(g),

where the union is taken over all -dimensional facets of .

Therefore,

Since the induced Riemannian metric on is a product metric of the flat metric on and the round metric on , we get

We claim that is a closed hemisphere in . To establish this, first observe that contains the vector with the only non-zero entry in the -st coordinate, and moreover, . If , there is nothing to prove further, and we are done. Therefore, we can assume . Then contains the unit ‘equatorial’ sphere : it is given by all unit vectors which in the coordinate system specified above have first entries equal to zero. Finally, let be two distinct points in with , be the supporting hyperplanes corresponding to , respectively, and let be the intersection of the closed half-spaces with respect to and that contain (if it happens that , then we assume that the half-spaces are chosen to be the lower half-spaces with respect to the normal vectors of and ). Then for every the hyperplane through with the unit normal vector is supporting to and hence to (here stands for the norm in ). By construction, , and we conclude and for every . Applying this reasoning to and each , we obtain that contains the closed hemisphere of ; for convenience, call this hemisphere ‘upper’. Let us show that, in fact, is equal to this hemisphere.

Assume the contrary, and let be a point in the open lower hemisphere. Let be the big circle through and , and let and be a pair of points in (where the boundary is taken with respect to the topology in ). Since is not in , this pair exists; and since the whole closed upper hemisphere is in , this pair necessarily lies in the closed lower hemisphere. Moreover, since is in the open lower hemisphere, one of the points or can be chosen to lie in the open lower hemisphere as well (for instance, by picking a pair of points in that are closest, in the spherical metric, to ). Hence, and the arc of connecting to through is the unique minimizing geodesic segment between and . By our construction above, all points in this segment belong to . But this is a contradiction because . Therefore, is equal to a closed upper hemisphere, as was claimed.

The claim implies that , which together with (3.11) yields (3.10). ∎

###### Proof of Theorem A.

Again, by scale-invariance of the left-hand side of (1.5), we can assume . By Proposition 2.4, there exists a nested sequence of -concave polytopes approximating from inside. For each of the polytopes in the sequence we know that (1.5) holds true. By Proposition 2.1, the left-hand side of (1.5) depends continuously on the body. From this the inequality for follows by passing to the limit as in the inequality for . The inequality case is proven.

Now we do a careful treatment of the equality case for general -concave bodies. This is where Lemma 3.2 and the properties of our approximation will come into play.

Equality case. As usual, assume that we have equality for a triple , that is

 (k−j)Wi(K)+(i−k)Wj(K)+(j−i)Wk(K)=0,

and let be a sequence of -concave polytopes that approximate ; this sequence is given by Proposition 2.4.

Similar to the proof of the equality case in Proposition 3.1, if we have equality for the triple , then we must have equality for the triple , that is

 Ei+1(K)=Wi(K)−2Wi+1(K)+Wi+2(K)=0

(here we use the notation from the proof of Proposition 3.1).

We claim that if there exists an index such that for we have equality in (1.5) for the triple , then is necessarily a -sausage body. Indeed, if

 Wi(Pl0)−2Wi+1(Pl0)+Wi+2(Pl0)=0,

then is a -sausage body by Proposition 3.1. By Proposition 2.4, . Therefore, , and since is convex, it follows that is a -sausage body as well.

Let us show that such an index exists. Assume the contrary, that is