Concave Convex-Body Functionals
and Higher Order Moments-of-Inertia
by H. Hadwiger, Bern
Translated from the original German version (H. Hadwiger, “Konkave Eikörperfunktionale und
höhere Trägheitsmomente” Comment. Math. Helv. 30 (1956), 285-296) by M. De Graef and corrected
by M. Bockstaller. The word “Eikörper” has been translated as “c-body”.
A class < of convex bodies (c-bodies) P, Q, . . . in a k-dimensional Euclidean space R is
called convex, when for all P, Q ∈ < always follows αP × βQ ∈ <
[α, β ≥ 0, α + β = 1].
Here we interpret λP (with λ > 0) to be a c-body, obtained from P through a dilatation with
respect to a fixed origin O of the space R; P × Q refers to Minkowski addition. This property of
a class of c-bodies thus not only refers to the size and shape of the body, but also to its location
in space.1
A functional ϕ(P ) defined over a convex class of c-bodies < is called (in the Minkowski
sense) concave, when for two arbitrary (non-empty) c-bodies P, Q ∈ < the functional inequality
ϕ(αP × βQ) ≥ αϕ(P ) + βϕ(Q)
[α, β ≥ 0, α + β = 1]
(A)
is satisfied. While no invariant property of the functional ϕ is postulated a-priori, in the cases
treated here we will typically deal with motion-invariant functionals.
As most prominent example we select first the functional
ϕ(P ) = V (P )1/k
(B)
where V represents the volume. The classical Brunn-Minkowski theorem, which occupies a key
1
Examples of convex classes are: a) convex polyhedra, b) rotational c-bodies with identical axes that contain
the origin, c) c-bodies that are contained as parts of a fixed c-body, when the latter itself contains the origin.
1
position within the theory of convex bodies, states, as is well known, that this functional over
the class of all c-bodies is invariant and, in our sense, concave.
The statement proven by W. Fenchel and A. Alexandroff 2 represents an important extension,
in which for i = 0, . . . , k − 1 the functionals
ϕ(P ) = Wi (P )1/(k−i)
(C)
are, in the same sense, concave functionals. Here, Wi represents the i-th Minkowski functional
(Quermaßintegral).
Following a suggestion by G. Pólya3 , I will prove in the present note, among other things,
that
ϕ(P ) = I(P )1/(k+2)
(D)
is a concave functional, where I denotes the moment-of-inertia with respect to the center-ofmass of P (the polar moment-of-inertia).
In the same context I will also consider certain higher-order moments-of-inertia, which
are written as simplex quadrature integrals and contain the ordinary moment-of-inertia as their
simplest special case. For these moments, inequalities are obtained that express the extremal
properties of the ball. In these inequalities, the norm N plays a central role, in addition to
the volume V . The former is proportional to the so-called mean width4 and, hence, measures
the (linear) size of the c-body. Volume and norm represent, in essence, the first and last mass
numbers in the series of non-trivial Minkowski functionals, for which V = W0 and N =
kWk−1 .5
2
W.Fenchel, “Généralisation du théorème de Brunn et Minkowski concernant les corps convexes,” C. r. Acad.
Sci. Paris 203, 764-766 (1936); A. Alexandroff, “Neue Ungleichungen für die Mischvolumen konvexer Körper,” C.
r. Aca. Sci. URSS (N.S.) 14, 155-157 (1937).
3
Written correspondence, Summer 1954.
4
We have N = (kωk /2)b̄, where b̄ is the mean width. ωk represents the volume of the k-dimensional unit ball.
5
For the lowest dimensionalities, we have in particular: a)k = 1: N = b̄ = length, b) k = 2: N = π b̄ =
circumference, c) k = 3: N = 2π b̄ = integral of mean curvature.
2
Among all c-bodies with fixed norm, it is for the ball that any motion-invariant and concave
functional takes on its largest possible value. On this fact, which, along with the proof of concaveness of a functional, immediately produces the solution of the associated extremal problem,
is based the meaning of the discussed concept within the theory of general c-body functionals,
as is also emphasized elsewhere.6
The proof that, among all c-bodies of prescribed norm (mean width), the ball exhibits the
largest moment-of-inertia, posed unexpected difficulties7 ; the present paper is the result from
the efforts to conquer these difficulties. I am indebted to Mr. G. Pólya for several practical and
methodical ideas and tricks.8
I.
In this first part, I will provide first a brief summary of inequalities which are valid for the
moments-of-inertia, and which will be derived in the subsequent parts. I will also formulate
the extremal properties of the ball, i.e., that among all c-bodies of fixed norm the ball has the
largest moment-of-inertia, and among all c-bodies of fixed volume the ball has the smallest
moment-of-inertia.
1.1.
Let P be a c-body in the k-dimensional Euclidean space R, and let p represent a variable
point inside P . For points and their position vectors we will use the same symbols. The centerof-mass of P is denoted by s. The polar moment of inertia I of P is then defined by the integral
Z
I(P ) =
6
|s, p|2 dp,
(1)
H. Hadwiger, “Konkave Eikörperfunktionale,” Monatshefte für Mathematik, 59, 230–237 (1955).
In the even case (k = 2), the corresponding inequality is given by L4 − 32π 3 I ≥ 0, where L is the circumference of the convex domain; the, to our knowledge, only proof for this relation is due to G. Pólya and
G. Szegö, “Isoperimetric Inequalities in Mathematical Physics,” Princeton 1951, p. 10, 123-126 and employs
function-theoretical techniques. As “Unsolved Problem Nr. 1” (Elemente der Math. 9, 111 (1954)) one is faced
with the task to find a simple proof for the above inequality.
8
See also, G. Pólya, “More isoperimetric inequalities proved and conjectured,” Comment. Math. Helv. 29,
112-119, (1955).
7
3
where |s, p| is the length of the line from s to p, and dp is the space differential (point density)
of the variable point p. The integration is carried out over all locations of p in the c-body P .
Let, furthermore, K ◦ and K represent balls, which have the same volume and norm as P ,
respectively, such that
V (K ◦ ) = V (P );
N (K) = N (P ).
(2)
Then the following inequality exists:
I(K) ≥ I(P ) ≥ I(K ◦ ).
1.2.
(3)
Now we define a series of k + 1 higher-order moments-of-inertia In (n = 0, . . . , k)
through the integral expressions
I0 (P ) = 1,
In (P ) =
1
cn
(4a)
Z
Z
···
|s, p1 , . . . , pn |2 dp1 . . . dpn
[1 ≤ n ≤ k].
(4b)
Here |s, p1 , . . . , pn | is the volume of an n-dimensional simplex, with vertices at the centerof-mass s and in each of the n variable points pi (i, 1, . . . , n) in the c-body P . The n-fold
integration spans all positions of the points pi in the c-body P . The constant cn is given by
k
1
(4c)
cn =
n
n!k n
and guarantees an appropriate normalization.
Obviously we have I1 (P ) = I(P ), from which follows that the series of integral moments
defined in (4) generalizes the classical moments-of-inertia to a complete system.
As a generalization of inequality (3) we have
I(K) ≥ I1 (P ) ≥ I2 (P )1/2 ≥ · · · ≥ Ik (P )1/k ≥ I(K ◦ ).
(5)
Furthermore, we have the following inequality, cyclic-symmetric in its three indices
Ia (P )(b−c) Ib (P )(c−a) Ic (P )(a−b) ≥ 1
4
[0 ≤ a < b < c ≤ k],
(6)
which expresses that In (P ) is logarithmically concave in its dependence on the integer variable
n.
1.3.
Through an obvious variation of the ansatz we obtain a further series of k +1 higher-order
integral moments Jn (n = 0, 1, . . . , k), defined by
J0 (P ) = V (P ),
Z
Z
1
Jn (P ) =
· · · |p0 , p1 , . . . , pn |2 dp0 dp1 . . . dpn
cn
(7a)
[1 ≤ n ≤ k].
(7b)
These are simplex quadrature integrals analogous to the previously defined In , where, however,
all n + 1 vertices pi (i, 0, 1, . . . , n) of the n-dimensional simplex should vary within the c-body
P.
It is apparent that these Jn do not represent new c-body functionals, but that they can be
reduced in a simple manner to the In . They can, hence, also be used as valid higher-order
moments-of-inertia. There is, in fact, the relation
Jn (P ) = (n + 1)V (P )In (P ),
(8)
which permits one to directly transfer all results for the moments In to the moments Jn . Those
we will obviously omit here.
II.
This second part is dedicated to the proof of concaveness of the regular polar and planar
moments-of-inertia.
2.1.
Let O denote the origin of the space and E a (k − 1)-dimensional plane containing O; u
is a normalized direction vector, orthogonal to E and starting in O. All vectors x, whose scalar
product with u are limited by (x, u) ≥ 0 and (x, u) ≤ 0, resp., mark the two (closed) halfspaces H+ and H− , resp., produced by E. In the following, we consider only c-bodies which
5
lie completely in the positive half-space H+ ; these bodies construct a convex class of c-bodies
<+ (see introduction) relative to the origin O as the center of dilatation.
The planar moment-of-inertia of P relative to E is given by the integral9
Z
|E, p|2 dp,
T+ (P ) =
(8)
where |E, p| is the distance between E and the variable point p inside P . The accompanying
function
ϕ(P ) = T+ (P )1/(k+2)
(9)
is defined over the class <+ , is definite, monotonous and linear by dilatation, so that we also
have
ϕ(P ) ≥ 0,
ϕ(P ) ≥ ϕ(Q)
(P ⊃ Q),
ϕ(λP ) = λϕ(P ).
Now we show that this functional is also (in the Minkowski sense) concave, so that relation (A)
of the introduction is satisfied.
Indeed: We embed the k-dimensional space R in a (k + 2) dimensional space R∗ , and
introduce two additional direction vectors v and w of R∗ , which are perpendicular to each other
and to R, and, moreover, are normalized. To a c-body P ⊂ R of the class <+ we assign a c-body
P ∗ ⊂ R∗ , whose points p∗ follow from the points p of P by the parameter representation
p∗ = p + (p, u)(ρv + σw),
(0 ≤ ρ ≤ 1, 0 ≤ σ ≤ 1).
One confirms easily that, for P, Q ∈ <+ and α, β ≥ 0, the relation (αP ×βQ)∗ = αP ∗ ×βQ∗ is
satisfied, where on the left side the Minkowski addition is carried out in R and on the right side
in R∗ . Furthermore it is to be seen immediately that, for P ∈ <+ , the identity T+ (P ) = V ∗ (P ∗ )
is valid, where V ∗ is the volume in the space R∗ . With the permutation relation above follows
9
Note from the translator: there is an error in the equation numbering in the original paper; there are two
equations (8). For consistency, we will “make” the same error.” Actually, we will make it twice, since now the
numbering of footnotes has been altered as well...
6
that the functional inequality (A) exists for the functional (9), when one employs the BrunnMinkowski theorem in the space R∗ .
2.2.
Let P be a c-body in the space R, which we imagine to be translated so that its center-of-
mass s coincides with the origin O. The two half-spaces H+ and H− introduced at the beginning
of the previous section divide P into the partial bodies P+ = P ∩ H+ and P− = P ∩ H− .
Since our definition for the planar moment-of-inertia of P fails, we introduce the two singlesided moments-of-inertia T+ (P ) and T− (P ) through the definitions T+ (P ) = T+ (P+ ) and
T− (P ) = T− (P− ). The sum T (P ) = T+ (P ) + T− (P ) is the usual planar moment-ofinertia T
of the c-body P with respect to the plane E, which is represented by the integral:
Z
|E, p|2 dp.
T (P ) =
(10)
With the Ansatz
ψ(P ) = T (P )1/(k+2)
(11)
we construct a functional defined over the class of all c-bodies which is, as is functional (9),
definite, monotonous and linear by dilatation; however, this functional also turns out to be
translation-invariant, so that ψ(P ) = ψ(Q) is valid whenever P and Q are equal with respect to
a translation (translationsgleich). Here it is, of course, understood that the plane E is fixed and
contains the origin. We wish to prove, next, that ψ is also concave (in the Minkowski sense).
Indeed: Let P and Q be two c-bodies in the spaces R and S = αP × βQ (α, β ≥ 0, α + β =
1) represents a linear combination; the center-of-mass s of S may coincide with the origin O.
Once again, we represent by u a normalized direction vector normal to E. Now we consider
new bodies Pτ and Qτ which are derived from P and Q by means of oppositely operating
transformations; in particular, let p and q be points of P and Q, then
pτ = p + τ βu
and qτ = q − τ αu
7
represent points of Pτ and Qτ . As one can easily show, for all τ we have the relation S =
αPτ × βQτ as before.
Next, we introduce the auxiliary functions ξ(τ ) = T+ (Pτ )/T− (Pτ ) and η(τ ) = T+ (Qτ )/
T− (Qτ ), where ξ(τ ) is defined in an open interval ρ < τ < ρ̄ and is there monotonous from 0
to ∞, whereas η(τ ) is defined in another open interval σ < τ < σ̄ but decreases monotonously
from ∞ to 0. Since for sufficiently small τ certainly Pτ ⊂ H− and Qτ ⊂ H+ is valid and
for sufficiently large τ , on the other hand, the opposite holds Pτ ⊂ H+ and Qτ ⊂ H− , since,
furthermore, both c-bodies never lie in the same half-space; since, hence, also S must be entirely in one half-space, which is, due to the assumption about the center-of-mass, impossible,
therefore there exist parameter values such that Pτ and Qτ are simultaneously divided by E in
two proper partial bodies. This means that the two definition intervals for ξ(τ ) and η(τ ) must
have a non-empty intersection.
Employing the monotony and continuity properties we find easily that a τ0 exists for which
ξ(τ0 ) = η(τ0 ) = ζ. Without loss of generality one may assume (translational invariance of the
functional!) that τ0 = 0, which is equivalent to the fact that the two originally selected bodies
P = P0 and Q = Q0 are located in the position indicated by the equality of ξ and η.
Next we consider that
S+ ⊃ αP+ × βQ+
and S− ⊃ αP− × βQ− .
From these we find, along with the already proven concaveness of the single-sided functionals
1/(k+2)
T+
1/(k+2)
and T−
, the inequalities
T± (S)1/(k+2) ≥ αT± (P )1/(k+2) + βT± (P )1/(k+2) .
However, by construction we have T+ (P ) = ζT− (P ) and T+ (Q) = ζT− (Q). From the two
relations above and by application of the inequalities
T+ (P ) + T− (P ) ≥ T (P )
and
8
T+ (Q) + T− (Q) ≥ T (Q)
(considering that the plane E does not necessarily go through the centers-of-mass of P and Q)
one obtains quite easily the desired formula, namely
T (S)1/(k+2) ≥ αT (P )1/(k+2) + βT (Q)1/(k+2) ,
and therefore the existence of inequality (A) for functional (11).
2.3.
Next we consider k pairwise orthogonal normalized direction vectors ui (i = 1, . . . , k)
and we place the k corresponding planes Ei (i = 1, . . . , k) through the origin O. We imagine
again a c-body in the space R to be translated so that its center-of-mass s coincides with O.
The sum of the k usual planar moments-of-inertia Ti (i = 1, . . . , k) of the body P with respect
to the planes Ei is then, as is well known, equal to the polar moment-of-inertia I, so that the
P
summation formula I = Ti can be noted. We remember that I was introduced via the integral
(1).
The ansatz
χ(P ) = I(P )1/(k+2)
(12)
introduces a functional that is defined for the class of all c-bodies; it is also definite, monotonous
and linear by dilatation, as is functional (11); but it is not only translation invariant but even
motion-invariant, so that χ(P ) = χ(Q) is valid when P and Q are congruent.
Now we wish to show that χ is also concave (in Minkowski’s sense). To establish the
existence of the functional inequality (A) it is sufficient to show that from I(P ) = I(Q) = 1
and S = αP × βQ (α, β ≥ 0, α + β = 1) the inequality I(S) ≥ 1 can be derived.
This is indeed the case: Let u be a variable direction vector and let T (P, u) indicate the
planar moment of the c-body P with respect to a plane that is orthogonal to u and goes through
the center-of-mass of P . Obviously T (P, u) is a continuous function of u. Next we consider the
continuous direction function f (u) = T (P, u) − T (Q, u). According to an auxiliary theorem
9
about continuous functions on spheres there exist k pairwise orthogonal directions ui (i =
P
1, . . . , k) so that f (u1 ) = . . . = f (uk ).9 Since
f (ui ) = I(P ) − I(Q) = 0 (considering
the summation formula for the moments and the validity of a theorem to be proven later), one
concludes that f (ui ) = 0 (i = 1, . . . , k) and, hence, that T (P, ui ) = T (Q, ui ) (i = 1, . . . , k).
From the already established concaveness of the functional T 1/(k+2) follow then the k inequalities
T (S, ui )1/(k+2) ≥ αT (P, ui )1/(k+2) + βT (Q, ui )1/(k+2) ,
which result, after using the above obtained correspondence of the individual moments of P
and Q, in the inequalities T (S, ui ) ≥ T (P, ui ). With the summation formula this results in
I(S) ≥ 1, which was to be proven.
III. In this third part we want to demonstrate the considered higher moments of inertia can be
represented as elementary symmetric functions of the planar major moments-of-inertia. This
facilitates expression of a mutual relationship between the simplex integrals in algebraic terms.
3.1.
We start once again from a c-body in the space R along with k directions ui (i = 1, . . . , k)
which intersect the center-of-mass of P and correspond to its principal moment-of-inertia axes.
The planar moments-of-inertia of P with respect to the normal planes are then the planar principal moments-of-inertia Ti (i = 1, . . . , k) with respect to the principal planes Ei .
As we will prove subsequently, the following representations are valid for the in (4) and (7)
defined higher order moments-of-inertia:
In (P ) =
1 X
(T1 , . . . , Tk )
n!cn n
9
(13)
This concerns the k-dimensional generalization of a theorem by S. Kakutani (k = 3). See also H. Yamabe and
Z. Yujobo, “On the continuous function defined on a sphere,” Osaka Math. J., 2, 19-22 (1950).
10
and
Jn (P ) =
where
P
n
n+1 X
V
(T1 , . . . , Tk ),
n!cn
n
(14)
(T1 , . . . , Tk ) are the elementary symmetrical functions of n-th degree of the k planar
principal moments-of-inertia Ti . In particular is thus I0 = 1, I1 = I = T1 + . . . + Tk , Ik =
k k T1 . . . Tk and J0 = V , J1 = 2V (T1 + . . . + Tk ), Jk = (k + 1)k k V T1 . . . Tk .
3.2.
The results (13) and (14) can be obtained by direct computation. Let p0 be a point in the c-
body P , which we wish to consider to be fixed, whereas an additional n points pi (i = 1, . . . , n)
are variable. Putting qi = pi − p0 (i = 1, . . . , n), then we have
|p0 , p1 , . . . , pn |2 = (
1 2
) det ||(qν , qµ )||,
n!
(ν, µ = 1, . . . , n).
Now we select the principal axes of inertia through the origin O, which itself should coincide
with the center-of-mass, as coordinate axes.
We denote the coordinates of the point pi by xiλ (i = 1, . . . , n; λ = 1, . . . , k), those of p0
by yλ (λ = 1, . . . , k), and we insert the expression
(qν , qµ ) =
k
X
(xνλ − yλ )(xµλ − yλ )
λ=1
for the scalar product into the above determinant! Then we integrate over the n variable points
pi (i = 1, . . . , n) inside the body P . After a few appropriate regroupings the following formula
results:
Z
Z
···
k
k
X
1 2X
|p0 , p1 , . . . , pn | dp1 . . . dpn = ( )
···
∆[λ1 , . . . , λn ],
n! λ =1
λ =1
2
1
n
0
where ∆[λ1 , . . . , λn ] = det ||Dλ0 i λj ||, (i, j = 1, . . . , n) and Dλµ
indicates the moment (the point
index is omitted here since it is not needed)
0
Dλµ
Z
=
(xλ − yλ )(xµ − yµ ) dp.
11
Defining
Z
Dλµ =
xλ xµ dp,
one obtains, in view of the vanishing of the static moments
R
xλ dp, the relation
0
= Dλµ + yλ yµ V.
Dλµ
Next we wish to compute the determinants ∆.
1st case : The λi are all different from each other. Then we have Dλi λj = 0 for i 6= j, since
this concerns a deviatoric moment with respect to the principal planes of inertia. The
evaluation of the determinant ∆ gives
∆[λ1 , . . . , λn ] = Tλ1 , . . . , Tλn 1 + yλ21 V /Tλ1 + · · · + yλ2n V /Tλn ,
where it is to be noted that Dλi λi = Tλi .
2nd case : The λi are not all different from each other. Since the determinant contains equal rows,
we have
∆[λ1 , . . . , λn ] = 0.
Now we let the point p0 coincide with the center-of-mass, or, equivalently, the origin O, so
that yi = 0 (i = 1, . . . , k), which results, using the equation for the first case,
∆[λ1 , . . . , λn ] = Tλ1 , . . . , Tλn ,
in the representation
Z
Z
···
|p0 , p1 , . . . , pn |2 dp1 . . . dpn =
1 X
(T1 , . . . , Tk ).
n! n
If p0 is also considered to be variable, then, using a formula which again follows from the first
case above
Z
∆[λ1 , . . . , λn ] dp0 = (n + 1)V Tλ1 , . . . , Tλn ,
12
we obtain in a completely analogous way
Z
Z
n+1 X
· · · |p0 , p1 , . . . , pn |2 dp0 dp1 . . . dpn =
V
(T1 , . . . , Tk ).
n!
n
Therefore, representations (13) and (14) have been achieved.
IV.
In this final part we prove the inequalities that were summarized in part I.
4.1.
Next we provide the proof for the cyclic-symmetric inequality (6). If we put
X
k
sn = 1/
(T1 , . . . , Tk ),
n n
then, according to a classic inequality for elementary symmetric functions,10 we have
sn−1 sn+1 ≤ s2n
[0 < n < k].
This inequality expresses that log sn is concave as a function of the integer-valued variable n.
In view of this circumstance one can verify directly (by taking the logarithm) the relation
a−b
sab−c sc−a
≥1
b sc
[0 ≤ a < b < c ≤ k].
Because of (13) and (4c) follows that sn = k n In . Substituting this in the inequality above, one
obtains (6). The equality sign holds if and only if all Tn have the same value, i.e., when the
c-body is one of “constant inertia”, in particular, therefore, when it concerns a ball.11
Putting a = 0, b = n, c = n + 1 in (6), we obtain, using (4a), the relation
1/(n+1)
In1/n ≥ In+1
and, hence, the proof for the central portion of the chain of inequalities (5).
10
See also G. Hardy, J.E. Littlewood and G. Pólya, “Inequalities,” Cambridge, 1934, p. 52, Theorem 51.
A body of constant moment-of-inertia has the property that its ellipsoid of inertia is a ball; as a consequence,
its planar and axial
p moments of inertia in all directions are equal to one another. The cylinder with radius R = 1
and height H = 12/(k + 1) is a body of constant inertia.
11
13
4.2.
Next we prove the left-most inequality of the chain of inequalities (5). We consider a
c-body Q, which arises from P through a Minkowskian rotational symmetrization, hence of the
form
Q = λ 1 P1 × λ 2 P 2 × · · · × λ m P m
[λi > 0, λ1 + · · · + λm = 1]
where the bodies Pi are all obtained from P through rotations. Using the concaveness of the
functional I 1/(k+2) and the rotational invariance of I, one concludes that I(Q) ≥ I(P ). The
norm N remains invariant under the rotational symmetrization, so that N (Q) = N (P ).
According to an auxiliary theorem that has already been proven elsewhere12 one can now
approximate a ball with arbitrary accuracy through an appropriate rotational symmetrization.
Hence, there exists, for an arbitrary > 0, a Q, so that d(Q, K) < , with K a ball with the
same norm as P and d the c-body distance. Employing the usual continuity arguments it follows
immediately that I(K) ≥ I(P ), which needed to be proven.
4.3.
Finally, we turn to the right-most inequality of the chain of inequalities (5). From the
definition (4) follows directly that the functional ∆ = Ik V −(k+2) is an affine invariant. Replacing Ik according to formula (13) by the planar principal moments of inertia, one obtains the
affine invariant functional ∆ = k k T1 , . . . , Tk V −(k+2) . However, we can transform the c-body P
through a simple affine transformation into a c-body P0 of constant inertia, for which all planar
principal moments-of-inertia have the same value T0 ; since kT0 = I0 we obtain the relation
∆ = ∆0 = I0k V −(k+2) .
R
We have I0 = r2 dp, where r represents the distance of the in P0 variable point p to
the center-of-mass. If dΩ represents the direction density (the surface differential of the (k −
1)-dimensional direction sphere), one can write dp = rk−1 drdΩ, and one arrives at I0 =
RR k+1
r drdΩ. In this integral we integrate, for a fixed direction, with respect to r and ob12
H. Hadwiger, “Altes und Neues über konvexe Körper,” Birkhäuser Basel 1955, p. 27. (The proof there is only
carried out in three dimensions.)
14
R
tain I0 = 1/(k + 2) Rk+2 dΩ, where R is the length of the line section enclosed in P0 of a
half-line originating in the center of mass and in a chosen direction. A comparison with the
R
integral relation V0 = 1/k Rk dΩ leads, upon application of a well-known inequality for integral averages of exponential arguments13 and knowing the directional integral dΩ = kωk , to the
(known) inequality
I0 ≥
k
−2/k (k+2)/k
ωk V0
.
k+2
We employ this partial result in the above representation of the functional ∆ and obtain ∆ ≥
ωk−2 [k/(k + 2)]k . Therefore the inequality Ik ≥ ωk−2 [k/(k + 2)]k V k+2 follows, and this is
equivalent to Ik (P )1/k ≥ I(K ◦ ), where K ◦ is a ball with the same volume as P . This concludes
the proof.
13
Loc. cit. Footnote 10, p. 143, Theorem 192.
15