Bull. London Math. Soc. 38 (2006) 839–846
C 2006 London Mathematical Society
doi:10.1112/S0024609306018728
THE FLOATING BODY PROBLEM
A. STANCU
Abstract
Let K be a convex body with boundary of class C 4 . We prove that, for a sufficiently small δ, the
floating body Kδ is homothetic to K if and only if K is an ellipsoid. The proof relies on properties
of the affine curvature flow.
1. Introduction
Initially a notion of differential geometry, the affine surface area was introduced
by Blaschke [3] for convex bodies in R3 with sufficiently smooth boundary. It was
later extended by Leichtweiss [6] to convex bodies in Rn+1 with sufficient boundary
smoothness as the quantity
Kp1/(n+2) dµ(p),
Ω(K) =
∂K
where Kp is the Gauss–Kronecker curvature at p ∈ ∂K and µ is the surface area
measure of K.
As it contains information on the boundary structure of a convex body, the
affine surface area arises naturally in questions regarding the approximations of
convex bodies by polytopes (see the survey by Gruber [5] and the papers of Schütt
[21] and Ludwig [7], to cite just a few). Equally important is the occurrence of the
affine surface area in isoperimetric inequalities of affine geometry, like the curvature
image inequality, the Blaschke–Santaló inequality, or Petty’s geominimal surface
area inequality [8, 11, 15]. A significant use of these inequalities in obtaining a priori
estimates for a certain class of non-linear PDEs was shown by Lutwak and Oliker
[10].
Given its implications in a series of essential problems, it was crucial to extend
the affine surface area to arbitrary convex bodies in Rn+1 without any smoothness
assumption on the boundary. Within the last decade or so, different (but equivalent)
extensions were given by Leichtweiss [6], Lutwak [9], Meyer and Werner
[14], Schmuckenschläger [18], Schütt and Werner [22], and Werner [25]. The
extensions of affine surface area to an arbitrary convex body K ⊂ Rn+1 in [6],
[14], [18], [22] and [25] have in common a geometric construction of a family of
convex bodies {Kδ }δ>0 related to the convex body K. This family is different in
each of the extensions. In [6] and [22], {Kδ }δ>0 denotes, respectively, the family
of floating bodies and the family of convex floating bodies, while in [14] it is the
family of Santaló regions, in [18] it involves convolution bodies, and [25] employs
illumination bodies. The affine surface area is then defined using the difference
between the volumes of K and Kδ , respectively.
Received 17 November 2004; revised 1 September 2005.
2000 Mathematics Subject Classification 52A20 (primary), 52A38 (secondary).
840
a. stancu
The definition of a floating body goes back to Dupin in 1822; see [4]. If K is a
convex body in Rn+1 , its floating body, Kδ , is the intersection of all halfspaces whose
hyperplanes cut off from K a set of volume δ. Blaschke [3], and later Leichtweiss
[6], were the first to show a relationship between floating bodies and affine surface
area. However, the study of floating bodies has been revived by work of Schütt and
Werner [20, 22, 23], partly because they extended the notion of affine surface area
to convex hypersurfaces without differentiability assumptions as mentioned earlier,
and partly because of the intrinsic interest of floating bodies. It is worth noting
that these bodies are relevant to other topics as well; see [2, 12, 13, 16, 20, 24],
to name just a few.
Our main theorem relates to a question on floating bodies. It can also be regarded
as yet another characterization of ellipsoids. Its statement is as follows.
Theorem 1. Let K ⊂ Rn+1 be a convex body with boundary of class C 4 .
There exists a positive number δ(K) such that Kδ is homothetic to K, for some
δ < δ(K), if and only if K is an ellipsoid.
If K is an ellipsoid, the implication is trivial. The question of whether the
condition ‘K homothetic to Kδ , for some δ’ implies that K is an ellipsoid has
been posed by Schütt and Werner. They have shown that if there is a sequence
{δk }k∈N 0 such that for all k ∈ N the floating body Kδk is homothetic to K with
respect to the same center of homothety, then K is an ellipsoid [23].
It is not surprising that the regularity of the boundary is an essential hypothesis
here. A central part of the Schütt–Werner result in [23] lies in showing that the
existence of a sequence of homothetic floating bodies implies that the boundary of
K is of class C 2 . We need the condition that the boundary of K is of, at least,
class C 4 , due to the techniques employed.
We end by noting that a key ingredient of our proof is the leading δ-term in the
asymptotic description of the affine area of Kδ .
2. Proofs and additional remarks
Let K be a strictly convex, compact body in Rn+1 with boundary of class C 4 .
We will see shortly that having K strictly convex is implicitly guaranteed by the
hypothesis of Theorem 1. Denote by h : Sn → R the support function of K, h(u) =
maxx∈K x, u . Let Kδ be the δ-floating body of K, where, for technical reasons,
we consider δ as δ = t(n+2)/2 for some t. Let ht : Sn → R be the support function
of Kδ .
Let u ∈ Sn be a fixed unitary direction. There exists a unique hyperplane of
normal u supporting the boundary of K:
Hu = {y ∈ Rn+1 | u · y = h(u)}.
(2.1)
Let Hu,t be the unique hyperplane parallel to Hu such that
V ({y ∈ K | ht (u) u · y h(u)}) = t(n+2)/2 =: δ;
thus
Kδ =
u∈Sn
{y ∈ Rn+1 | u · y ht (u)}.
(2.2)
(2.3)
the floating body problem
841
Choose coordinates x1 , x2 , . . . , xn+1 in Rn+1 such that {e1 , . . . , en , u} is a basis
of Rn+1 and the supporting point, {p} := Hu ∩ ∂K, lies at the origin. Then ∂K is
locally a graph in these coordinates,
n
1 xn+1 = −
hij xi xj + O(|x|3 ),
(2.4)
2 i,j=1
where hij is the second fundamental form of ∂K at the supporting point. Moreover,
there exists a volume-preserving linear transformation that fixes u and brings ∂K
locally to the form
n
1
xn+1 = − Kp1/n
x2i + O(|x|3 ),
(2.5)
2
i=1
where Kp = det[(hij )ij ] is the Gauss–Kronecker curvature of ∂K at p, which can
be viewed as a function on Sn , namely Kp (u).
Thus, if d denotes the distance between the hyperplanes Hu and Hu,t , one has a
description of δ = t(n+2)/2 , the cut-off volume, as
V ({y ∈ K | h(u) − d u · y h(u)})
= 2n/2 ωn Kp−1/2 (u)d(n+2)/2 + O d(n+3)/2 , (2.6)
where ωn is the volume of the unit ball in Rn+1 .
This leads to a description of the support function of the floating body in terms
of the support function of the original convex body:
1/(n+2)
ht (u) = h(u) −
= h(u) −
n/(n+2)
Kp
n/(n+2)
t + O t3/2
O t3/2 ,
2/(n+2)
2
ωn
c(n)Kp1/(n+2) (u)t +
(2.7)
2/(n+2) −1
ωn
)
is a constant depending on the dimension.
where c(n) := (2
Consider now the hypothesis that K is homothetic to Kt . Consequently, there
exists λ > 0 such that for all u, one has ht (u) = λh(u). Note also that K could be
assumed to be strictly convex due to a fact proved by Schütt and Werner, which
states that the floating body of a convex body is strictly convex [23]. The hypothesis
that K is homothetic to Kt implies that K itself is strictly convex.
We choose the ratio of affine areas to describe λ. Recall that the affine area of a
convex body with smooth boundary is
1/(n+2)
Ω(K) =
Kp
dµ =
Kp−(n+1)/(n+2) dµSn ,
(2.8)
∂K
Sn
where dµ is the surface area of K and dµSn is the surface area of Sn . Hence
(n+2)/n(n+1)
Ω(Kt )
λ=
.
(2.9)
Ω(K)
To obtain an asymptotic expansion for λ, we use the following proposition. As
regards notation, we follow [19] so that s(K1 , K2 , . . . , Kn , u) stands for the mixed
curvature function of K1 , K2 , . . . , Kn ; thus
1
V (K, K1 , K2 , . . . , Kn ) =
hs(K1 , K2 , . . . , Kn ) dµSn,
n + 1 Sn
where V (K, K1 , . . . , Kn ) here and throughout denotes the mixed volume. Therefore
s can be viewed as an n-linear symmetric functional on support functions. In the
842
a. stancu
next proposition we consider the extension of s on all smooth functions on the unit
sphere Sn .
Proposition 2. If Kδ denotes the δ-floating body of a strictly convex body K,
then
Ω(Kδ ) − Ω(K)
= −C(n)
vs(v, h, h, . . . , h) dµSn,
(2.10)
lim
δ→0
δ 2/(n+2)
Sn
where
v(u) := Kp1/(n+2) : Sn → R+
and
(n + 1)n
c(n)
C(n) =
n+2
is a constant depending only on the dimension.
Proof.
Regarding t as a varying parameter, we note that (2.7) implies that
lim
t→0
ht (u) − h(u)
= −c(n)Kp1/(n+2) (u).
t
(2.11)
d
(ht (u)) = −c(n)Kp1/(n+2) (u),
dt
(2.12)
This can be viewed as
which is the scalar form of the affine curvature flow studied previously by Sapiro and
Tanenbaum (n = 1; see [17]), and by Andrews (n > 1; see [1]). This explains the
earlier notation for v as the inward velocity of convex hypersurfaces moving under
the affine curvature flow.
Let K evolve by the affine curvature flow, and consider the evolution equation of
the affine surface area of K. One has
−(n+1)/(n+2)
Ω(K) =
Kp
dµSn =
s[h, h, . . . , h](n+1)/(n+2) dµSn,
(2.13)
Sn
Sn
where s[h, . . . , h] is the curvature function of K. Under sufficient regularity
assumptions on ∂K, we see that s[h, . . . , h] is the reciprocal of the Gauss curvature
as a function on the unitary directions of Sn . Thus
n+1
d
(Ω(K)) = −
n
c(n)s[h, h, . . . , h]−1/(n+2) s[v, h, . . . h] dµSn
dt
n + 2 Sn
n+1
nc(n)
=−
Kp1/(n+2) s[v, h, . . . h] dµSn
n+2
n
S
= −C(n)
vs[v, h, . . . h] dµSn.
(2.14)
Sn
This is equivalent to
Ω Kt(n +2)/ 2 − Ω(K)
= −C(n)
lim
vs[v, h, . . . h] dµSn,
t→0
t
Sn
where t = δ 2/(n+2) , thus concluding the proof of the proposition.
(2.15)
843
the floating body problem
Remark 1. Similarly, we can show that
V (Kδ ) − V (K)
=
−c(n)
vs(h, h, h, . . . , h) dµSn
lim
δ→0
δ 2/(n+2)
Sn
= −c(n)Ω(K),
(2.16)
which was known to Blaschke [3] for n = 2, and was extended to n > 2 by Schütt
and Werner [22].
Remark 2. We state here two important properties of the affine curvature flow
(2.12). These are due to Andrews [1], and to Sapiro and Tannenbaum [17] in the
case n = 1. Ellipsoids are the only convex bodies that evolve homothetically under
the affine curvature flow. Any convex body deformed by the affine curvature flow
will shrink to a point in finite time T , and for any sequence of times {tn }n T ,
there exists a subsequence (denoted for simplicity in the same way) such that the
{K(tn )}n , normalized so that they enclose a constant volume, converge in the C ∞
norm to an ellipsoid.
Lemma 3. If K is homothetic to Kδ for δ (and thus t) sufficiently small, then
1/(n+2)
Kp
: Sn → R+ (or, equivalently, v) is the support function of a convex
body K in Rn+1 .
Proof.
Using Proposition 2, one obtains
(n+2)/n(n+1)
Ω(Kt )
λ=
Ω(K)
⎛
⎞(n+2)/n(n+1)
n
vs[v,
h,
.
.
.
h]
dµ
S
⎜
⎟
Sn
t + O(t2 )⎟
=⎜
⎝1 − C(n)
⎠
Ω(K)
= 1 − c(n)
Sn
vs[v, h, . . . , h] dµSn
Ω(K)
and, furthermore, using (2.7),
h(u) =
Kp1/(n+2) (u)
t + O(t2 ),
(2.17)
Kp1/(n+2) dµ
·
+ O(t1/2 ).
∂K
(2.18)
vs[v, h, . . . , h] dµ
Sn
Sn
Note that this asymptotic expansion is different from the one obtained by Schütt
and Werner in [23], which can be derived by using (2.16).
Furthermore, formula (2.18) leads to
⎤n
⎡ 1/(n+2)
K
dµ
p
⎥
⎢
⎥ + O(t1/2 ),
∂K
sn (h(u)) = sn (Kp1/(n+2) (u)) · ⎢
(2.19)
⎦
⎣
vs[v, h, . . . , h] dµSn
Sn
where sn (h) := s(h, h, . . . , h) is the curvature function of K.
844
a. stancu
The left-hand side is 1/Kp , and thus is strictly positive. So is
vs[v, h, . . . , h]dµSn,
Sn
as Ω(K) decreases strictly under the affine curvature flow. Therefore, for t
sufficiently small, we have
sn (Kp1/(n+2) ) > 0,
and we apply the Minkowski existence theorem [19] to conclude the proof.
Remark 3. K is homothetic to K if and only if K is an ellipsoid.
If K and K have the same center of homothety, the claim is a direct consequence
of Petty’s lemma [15]. This states that if K is a convex body in Rn+1 with C 2
boundary such that, for some constant c, the equality
hn+2 (u) = cKp
holds for all u, then K is an ellipsoid. More generally, if K is homothetic to K,
then K evolves self-similarly under the affine curvature flow. By Remark 2, K is
an ellipsoid.
Using the interpretation of v as the support function of a convex body K ⊂ Rn+1 ,
we have another form of (2.18) in terms of mixed volumes:
h(u) = Kp1/(n+2) (u) ·
V (K, K, K, . . . , K)
+ O t1/2 .
V (K, K, K, . . . , K)
(2.20)
Suppose that K is not an ellipsoid. By the Brunn–Minkowski inequality, this implies
that
V (K, K, K, . . . , K)
(2.21)
h(u) > Kp1/(n+2) (u) ·
+ O t1/2 .
V (K, K, K, . . . , K)
Given the regularity of ∂K, equality is attained if and only if K is an ellipsoid.
Moreover, if δ (and thus t) is small enough, the inequality
h(u) > Kp1/(n+2) (u) ·
V (K, K, K, . . . , K)
V (K, K, K, . . . , K)
(2.22)
also holds, for all unitary directions u.
The proof of Theorem 1 is now concluded by the following result.
Proposition 4. If K is a compact convex body with smooth boundary,
containing the origin, then
1/(n+2)
maxn
u∈S
Kp
(u)V (K, K, . . . , K)
1,
h(u)V (K, K, K, . . . , K)
(2.23)
with equality if and only if K is an ellipsoid.
Proof.
Let
1/(n+2)
φ(u) =
(u)V (K, K, . . . , K)
Kp
.
h(u)V (K, K, K, . . . , K)
Note that φ is invariant under rescaling. Let K evolve by the affine curvature flow,
the floating body problem
845
and consider the evolution equation of ln(φ):
d
hs[1, h, . . . , h]dµSn
d
1 dv
1 dh
1 dΩ(K)
dt Sn
(ln(φ)) =
−
+
−
dt
v dt
h dt
Ω(K) dt
hs[1, h, . . . , h]dµSn
Sn
(n + 1)c(n)
vs[1, h, . . . , h]dµSn
v
nc(n)
Sn
Kp s[v, h, . . . , h] + c(n) −
=
n+2
h
hs[1, h, . . . , h]dµSn
Sn
n(n + 1)c(n)
vs[1, v, . . . , h]dµSn
Sn
+
.
(2.24)
(n + 2)
vs[1, h, . . . , h]dµSn
Sn
Moreover,
s[v, h, . . . , h] =
n
Hij (∇i ∇j v + δij v),
i,j=1
where the coefficients Hij depend only on h and form, at each point, a positive
definite matrix.
On the other hand,
V (K, K, . . . , K)
v = φh
V (K, K, K, . . . , K)
and thus
∇i ∇j v = [(∇i ∇j φ)h + 2∇i φ∇j h + φ(∇i ∇j h)]
V (K, K, . . . , K)
.
V (K, K, K, . . . , K)
At u where φmax is reached, one has
Kp s[v, h, . . . , h] V (K, K, . . . , K)
φmax
V (K, K, . . . , K)
and
V (K, K, . . . , K)
v
=
φmax .
h
V (K, K, . . . , K)
Standard parabolic equations theory implies that φmax satisfies the following
differential inequality:
1 d
2(n + 1) V (K, K, . . . , K)
(φmax ) c(n)
·
· (φmax − 1).
φmax dt
n+2
V (K, K, . . . , K)
(2.25)
Thus if φmax < 1, it will stay so for the whole time of the flow evolution, t T .
If K shrinks to a point other than the origin, then there exists a t0 < T such that
0 ∈ ∂K(t0 ). Thus, as t → t0 , h(t) 0 and φmax ∞, reaching a contradiction.
If K(T ) is the origin, the asymptotic behavior of the flow implies that φ(tn ) → 1.
However, this too is impossible if (2.25) holds.
It seems to us that results analogous to Theorem 1 can be obtained for other
pairs of bodies via similar methods. In particular, illumination bodies are good
candidates, as they too are invariant under affine transformation. However, in that
case, it is not the affine curvature flow whose properties should be used, but rather
a deformation that is technically more complex and has not been studied before,
since it does not preserve the convexity of evolving curves and hypersurfaces for all
time. We hope to address this problem in a further paper.
846
the floating body problem
Acknowledgements. I would like to thank Elisabeth Werner for very helpful
discussions on this problem, and Shlomo Reisner for very useful remarks.
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A. Stancu
Department of Mathematics
Polytechnic University
of New York
Brooklyn, NY 11201
USA
Département des Mathématiques
et de Statistique
Université de Montréal
Montréal, QC
H3C 3J7, Canada
astancu@duke.poly.edu
stancu@dms.umontreal.ca