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Geometriae Dedicata 55:319-340, 1995. 319 © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

On the Mean Number of Normals Through a Point in

the Interior of a Convex Body

DANIEL HUG Mathematisches Institut, Albert-Ludwigs-Universitiit, Albertstrafle 23b, D-79104 Freiburg i. Br., Germany

(Received: 30 December 1993; revised version: 4 May 1994)

Abstract. Recently, Kathy Hann established bounds on the average number of normals through a point in a convex body K, in the cases where K is either a polytope or sufficiently smooth. In addition, an Euler-type theorem was obtained for these particular classes of convex bodies. In the present work we show that all these statements are true for an arbitrary convex body K. For this purpose measure geometric tools and a general approximation technique will be essential.

Mathematics Subject Classifications (1991): Primary 52A40, 52A38; Secondary 53C65, 52A22.

1. Introduction and Statement of Results

The general setting throughout this paper will be the d-dimensional Euclidean space ]~a, with scalar product ( -, -) and norm II • II. We assume d >_ 2. The class of all compact, convex sets with nonempty interiors will be denoted by/~0 d, its members will be referred to as convex bodies. We define the support function h(K, .) of K • /~0 d by h(K, u) := max{(u, Y) IY • K} for all u • I~ d. Let K • / C d be a convex body and let x • bd K be a point in the boundary of K. Then the set of all vectors u • IR d such that (u, x) = h(K, u) makes up the cone N(K, x) of exterior normal vectors of K at x. Alternatively we could write

N(K , x ) : = {u • ~d \{o} I x • H(K, u)} U {o},

where H(K, u) := {y • ~d] (u, y) = h(K, u)} is the support plane of K with exterior normal vector u • I~d\{o}.

Let us fix p • ]~d and K • / ~ 0 d for the moment. As usual we set S ~-t := {x £ I~dl Ilxll = 1}. Now we can introduce the number n ( g , p) • [0, co] of normals x + ~ u that contain the point p, where z • b d K and u G N(K, x) fq S d-1. More precisely, if we define the generalized (unit) normal bundle of K by

A/'(K) := {(x, u) • bd K × s d - l l u • N(K, x)},

then

n(K, p) := card{(x, u) G A/'(K) I P G x + ~u} .

320 DANIEL HUG

To begin with, we report on some local properties of the function n(K, .). An obvious estimate is 2 _< n(K, p) <_ ~ for arbitrary K E/COd and p E ~d. No

better inequalities are available in general. This is already shown by the example of a ball. Heil [14, Th. 1] proved that there is at least one point p E int K (the interior of K) with n(K, p) > 6, if d > 3. For centrally symmetric K E/COd he sketches a proof that n(K, p) >_ 2d for a suitably chosen point p E int K. In contrast to the situation for the ball, an amazing result by Zamfirescu [24, Th.] implies that for most convex bodies K E /COd and for most points p E I~ d we even have n(K, p) = oc. In this context 'most ' has to be understood in the sense of Baire category. Recall that/COd is a Baire space in the topology which is induced by the Hausdorff metric. See, e.g., [12] for the details. This is, however, not a contradiction to our Corollary 1.2, saying that for an arbitrary convex body K E/C g we have n(K, p) < ~ for almost all p E ~d with respect to d-dimensional Lebesgue measure A d.

Another way to study normals of a convex body, which is of a global nature, is motivated by some work of Santal6 [18, pp. 533-534]. There it is suggested to find the limits between which the integral

± ( g ) := fK n (K, y) dAd(y)

varies. The problem of determining the behaviour of the motion invariant functional I : /cd _+ i~>0 := [0, oc] may also be reinterpreted as estimating the mean value (expectation, first moment) of the random variable n(K, .). For this to make sense we have to consider the completion of the measure space (K, ~3(K), Y(g)- l)~ d L K) as underlying probability space. In this situation ~3(K) denotes the a-algebra of Borel sets in K and V(K) := Ad(K) > O.

Only recently, Kathy Hann [13] was able to establish estimates for the average n(K) := I (K) /V ( K) , when K is either a polytope or bd K is a CZ-submanifold of ]i~ d with everywhere positive Gauss-Kronecker curvature (in the latter case, K is said to be of class C~). Unfortunately it is not possible to extend her results to the case of a general convex body via standard continuity arguments, since the functional n :/Co d ---+ ]~>0 actually is discontinuous. In fact, for regular 2m-gons P2m E ]C~ in the plane we obtain n(P2m) = 8 for all m E 1~ [13, p. 38], whereas obviously n(B2(o, 1)) = 2. We use Bd(x, r) to denote the closed Euclidean ball with centre x E I~ d and radius r > 0. Nevertheless, in the following we shall generalize the main estimates from [13] and at the same time unify the earlier approach.

THEOREM 1.1. Let K E /COd be an arbitrary convex body. Then the similarity >0 invariant functional n •/COd --+ ~ - satisfies the inequalities

2 < n (g) < V ( K + OK) - - V ( K ) - 1 , ( 1 )

where D K := K - K denotes the difference body of K.

NORMALS THROUGH A POINT IN A CONVEX BODY 321

The left inequality in (1) is sharp. This is shown by the example of a ball K = Bd(o, 1). If, in addition, K is centrally symmetric, the inequality can be expressed as

2 _< n ( K ) __ 3 d - 1. (2)

The example of the d-cube Cd shows that the right estimate is sharp in this restricted situation.

COROLLARY 1.2. For an arbitrary convex body 1( E IC d the number n( l ( , p) is finite for Ad-almost every point p C N d.

The other main result in [13] refers to an Euler-type identity which turned out to be useful in deriving estimates for the quantity n(K) . Again the statement in [13, (4.1.5) and (4.2.1)] was subject to severe regularity assumptions for the boundary of the convex body K. In order to be able to state a general version of this result, we first have to introduce some notations. For a convex body K C E d and (x, u) C Af(K) let

u) : = n I ( K n +

denote the length of the secant in K, generated by the line x + lRu. Here and in the following we denote by 7-/k, k >_ 0, the k-dimensional Hausdorff measure in ll~ d. This measure is independent of the dimension of the surrounding space. In particular we have A d = 7-/d in I~ d. For a definition and details see [9] or [17]. Next, for K C /C d we introduce the nearest-point map (metric projection) p(K, .) : ]R d ~ K , which assigns to an arbitrary x E ~d the uniquely determined po in tp (K , x) C K satisfying

IIx - p(K, x)ll = min{llx - yll ly I ( }

(see [20, 1.2 and Lemma 1.8.9]). A boundary point x E bd K is called regular, if d i m N ( K , x) = 1, and the set of regular boundary points of K is denoted by regK. In [20, 2.2, p. 78] the maping aK : reg K --+ S d-1 is defined by letting crK(x), x E reg K, be the unique unit outer normal vector of K at x. We extend this definition of a g by putting

x - p (K , x) aK(X) := [1 x _ p(K, x)ll

for x E ]Rd\K. Thus o" K is defined on reg K U (]Rd\K). If we define the parallel body K c, e > 0, as the Minkowski sum K + Bd(o, e), we get aK,(X) = 6rK(X)

for x E bd K ' = reg K ' . We also need the composed mapping

FK : ~ d \ K -+ 11~ d × S d-l , FK := (p(K, -), aK).

322 DANIEL HUG

For a Borel subset r 1 C ]~a × sd-1 and p > 0 we can define the local parallel set Mp(K, ~) by

Mp(K, ~1) := {x C ]~dl0 < IIx - p(K, x)ll _< p, FK(x) e 7}.

For these sets a generalized Steiner formula holds, i.e. a polynomial expansion of A d ( MR (K, r I ) ) in powers of p > 0, the coefficients of which are positive me asures Or(K, ~/) on the Borel a-algebra ~(~d × sd-1) for r E {0 , . . . , d - 1}:

Aa(Mp(K, q)) = _ S TM pd-r d Or(K, 7l). (3) d =o r

This is shown in [20, Th. 4.2.1, (4.2.4)], where further properties of these measures can be found.

We are now prepared to state the announced general Euler-type identity.

THEOREM 1.3. For an arbitrary convex body K C I(.~

( 1 - ( - 1 ) d ) V ( K )

= ~..~(-1)rct a(K, X, U) r+l d 0 a - l - r ( K , x, u). (4) r=o r + 1 (K)

In [13] this was proved in the cases where K is either a polytope or of differentiability class C 2. Relation (4) is a generalization of an identity, which was treated by Blaschke [4], Dinghas [7], Santai6 [19], Debmnner [6], and Chakerian and Groemer [5], in the special case of a convex body of constant width.

Although our results can be formulated entirely within convexity theory, we will have to apply tools from geometric measure theory. Part of the motivation for the present approach stems from work of Zfihle [22] and Kohlmann [15], [16]. Their work may be understood as an effort to transfer concepts and theorems which originally pertained to convex differential geometry to the more general setting of convexity theory. The measure geometric methods of Federer [8], [9], as well as the fundamental investigations of Aleksandrov [1 ], [2] and Fenchel and Jessen [10] have been essential for this process.

2. Preliminaries

In the present section we gather some material which appears at various places in the literature [21], [22], [15], [11], [16], but mainly in a more general guise than is needed for our purpose. We therefore streamline the current presentation to the Euclidean, convex situation, on which we are focusing in this paper. As regards proofs - the restricted situation would allow for considerable simplifications - we partially refer to the literature cited above for the details. For general terminology


with respect to geometric measure theory [9] is our reference, for convexity theory we follow [20].

In Section 1 we already defined mappings p(K, .), aK, and FK which are connected with an arbitrary convex body K E E0 d. Let 9t,: C ]~dXK be the common set of differentiability points for these three mappings, and let the coordinate projection 7rl : ~d × ~d __+ ~d be given by 7rl(X, y) := x. We then have for all e > 0

A/'(K) = FK(I~dXK) = F g ( b d KC).

Thus .Af(K) is a compact subset of]~ a ×/~d, and FK [ bd K ' : bd K ¢ --+ Af(K) is a bi-Lipschitz homeomorphism. This follows from the Lipschitz property of p(K, .) as well as from the representation of its inverse t¢ := (FK ]bd K~)-I :

t, N ( K ) --+ bd K', (z, ~) ~ • + , u

A more general investigation may be found in [21, §2 and Th. 4.3]. For each e > 0 one can show that 7-/d-l(bd K~X~DK) = 0 [21, Th. 3.3]. An essential step in the proof of this fact is the observation that p(K, .) is differentiable in x C ]RdXK if and only if p(K, .) is differentiable in p(K, x) + AaK(x) for an arbitrary A > 0. In addition, a symmetric, positive semidefinite bilinear form is defined for y E DK f) b d K ~ on the (d - 1)-dimensional linear space Ty bd K C = Tan(bd 1( ~, y) = at,:(y) l [9, §3.1.21] by

IIy(u, v) = (DaK(y)(u), v), u, v e Ty bd K ~.

For a proof the relation

V d / c ( y ) - y -p (K , y) d~(y) - '~ (Y) ' Y ~ RdXK'

may be used, where dK(y) := Ily - p(K, y)l]. The distance function dK is continuously differentiable on ~dXK [8, Th. 4.8, (3), (5)] and its second differential D2dK(y) exists exactly for y E 79ff. The asserted symmetry then follows from

a~(u, v ) = D2a~(y)(~, v)

together with [9, 3.1.11]. Since dK is convex for K E /~0 d, it follows that II v is positive semidefinite. Alternatively this can be viewed as a special case of [3, (4.6), (4.7), (4.9)]. See also [11, Prop. 3.5] or [16, §1, Lemmas 1.8 and 1.9].

In the following survey we denote by U l , . . . , ua-1 an orthonormal basis of eigenvectors with corresponding eigenvalues kl(y), . . . , kd-l(y) E [0, co) for

324 DANIEL HUG

II v, y C DK. (Here we deviate from the notation used in [15], [16].) Then the limit

ki(K, FK(y)) := ki(FK(y)) := lim ki(y) ~o 1 + (t - , )k~(y)

oo, if ki(y) = c -1

= k i ( Y ) £-1 ' 1 , eki(y)' if ki(y) <

i C { 1 , . . . , d - 1}, exists and depends only on FK(y) and not on e := dK(y). This was obtained in [22, §1] and [16, 2.3 Def. and Th.]. From this definition one can derive that for 7-/d-l-almost all (x, u) -- v E AF(K) and for y := t~(v) the relation

ki(K, v) 1 ki(y) = ki(K ~, x + eu, u) = 1 + eki(K, v) < -e < oo (5)

holds for i E {1 , . . . , d - 1}. Moreover, for such y E 79K CI b d K ' , e > 0, v = FK(y), an orthonormal basis of the (d - 1)-dimensional linear subspace Tand-l(7-[ d-1 , Af(K), v) C ~d × ~d of (7-/d-1 t. Af(K), d - 1) approximate tangent vectors at v is given by

( 1 -elei(y) .ui,

(1 - ,k~(y))2 + k~(y)2

k~(y) .u~)l x/(1 _ ,k~(y))2 + k~(y)2 i = l , . . . , d - 1 ) .

This is shown by an application of the tangential map

DFK(y) i Tan(bd K ~, y)" Tan(bd K ' , y)--+ Tand-l(7-I d-1 ..Af( K), v),

which proves that

T a n a - l ( ~ d-I L Af(K), v) = Tan(Af(K), v)

= im(DFK(Y) lTan(bd K ~, y))

according to [9, Lemma 3.2.17]. The right equality is also implied by [21, Prop. 4.2]. It will be convenient to use

1 - eki(y) 1 m 9

4 ( 1 - e k i ( y ) ) 2 + ki(y) 2 4 1 + k,(v) 2


and similarly

k4(y) k (v) ~//(1 - e k i ( y ) ) 2 + ki(y) 2 ~/1 + ki(v) 2

In order to verify these relations one has to distinguish the two cases ki(y) < e -1 and ki(y) = e -1. See also [22, §3, Proof of Prop.]. It is now easy to compute the following (7-/d- 1 t. Af(K), d - 1) approximate Jacobians for 7-/d-l-almost all (x, u) e N ( K ) :

ap Jd-l(Trl IX(K)) (x , u) = d-1 ~/1 ki(K, u) 2' II4=l + X,

(6)

d - 1

ap I I 4=1

1 + eki(K, x, u)

~/1 + ki(K, x, u) 2 (7)

Here we should recall [9, §3.2.16]. In the same way we derive for the mapping

f • Af(K) x [0, diam K] ~ Rd, (x, u, A) ~ x - Au,

for 7-/d-almost every (x, u, A) E N ' (K) × [0, diam K], the equation

d-1 11_ Ak4(K, x, u)[ apJdf(x , u, A ) = I I ~ l + - k / - ~ ,

4=1 X~ U) 2 (8)

We shall need these approximate Jacobians in the following in order to be able to apply Federer's general area/coarea formula [9, §3.2.22].

A comparison of the coefficients in Steiner formulae for local parallel sets, given in [20, Th. 4.2.1, (4.2.4)] respectively [22, Th. 2, (10)], and essentially also in [16, Th. 2.7], after an obvious extension of the reasoning there, leads finally to

d- 1) O d - l - r ( K , ~) T

= /Af El~_il<...<ir~d-1 ki,(K, v)...ki~(K, v) dT~d_l(v )

(K)nn I-I~i2_t ~/1 + ki(K, v) 2 (9)

for r E { 0 , . . . , d - 1} and Borel sets r/ C ~ d X S d-1 . If r = 0, the empty sum in the numerator has to be interpreted as 1. The connection between these generalized curvature measures O~(K, .) and Federer's [8] curvature measures

326 DANIEL HUG

Cr(K~ "), respectively the surface area measures Sr(K, .) of Aleksandrov [1] and Fenchel and Jessen [10], is described by the relations

Or(K, ~ x s d-~) = Cr (K, ~), /~ • ~(~d), (lO)

and

Or(K, ~d x ~) = St(K, oJ), w • ~3(sd-1), (11)

r • { 0 , . . . , d - 1}. From (5), (6), (9), (10), and also from [9, Th. 3.2.22], one derives for/3 • ~(IR d) and e > 0

( d- l ) Cd-l-r(KC, fl)= ~b r d K~r-I/3 Z

l<il<...<ir<d-1

x k i l ( K ~, x, aK, (X) ) . . . k i r (K ~, x, a i ( , ( x ) )d~d- l (x ) , (12)

where aK~(x) = aK(x) E N ( K ~, x) N S d-1 is uniquely determined for all x E bd K ~.

3. Proof of the Main Results

Let K E /(;0 d be a convex body and A/'(K) its corresponding generalized normal bundle. Instead of the cylindrical set M, considered by Harm [13] in the smooth case, we define M ( K ) C ~d X ~d X ~ by

M ( K ) := {(x, u, )Q I (x, u) • Af (g ) , )~ • [0, a(K, x, u)]}.

Therefore the first problem we encounter is to show the measurability of this newly defined set M (K) . A verification is provided in the course of the next two lemmas. The first of these will also be relevant to our proof of Theorem 1.3.

LEMMA 3.1. Let K • IC d be a convex body, (x, u) • Af (g) , and e > O. Then we have (x + eu, u) • N ' (K ~) and

lim a ( K ~, x + eu, u) = a ( K , x, u). (13) el0

In addition, the mapping

a ( K , . ) :Af (K) ---+ 1~>0, (x, u ) ~ a(K, x, u),

is Borel measurable.

NORMALS THROUGH A POINT IN A CONVEX BODY 3 2 7

Proof For each e > 0 the equality K ~ N (x + eu + ~u ) = K c O (x + ~u ) holds, and we have according to [20, Lemma 1.8.1]

l im[K ~ n (x + ~u)] = N [ K ~ n (x + ~u)] el0

e>0

= K N (x + ]~u).

Viewing K ~ N (x + I~u), e > 0, as convex bodies in x + ~u , the first statement is implied by [20, Th. 1.8.16]. To show the measurability statement, we observe that tc and

bd g ~ --+ ~>o, y ~ a ( g ~, y, aK~(y)),

are continuous maps. See [20, Th. 1.8.8] for the second map. Hence, the composi- tion of these two maps,

A/'(K) ~ ~>0, (x, u) ~-+ cr(g ~, x q- eu, u),

is continuous as well. According to the first part of the present proof, the mapping

a(K, .): Af(K) ~ ~>_o, (x, u) ~ a(K, x, u),

is obtained as limit of a sequence of continuous functions. Hence this limit is Borel measurable. []

LEMMA 3.2. The set M ( K ) C IR d x I~ ~ x 1R is closed and thus Borel measurable.

Proof For e > 0 define the approximating sets

M~(K) := {(x, u, A) I (x, u) e H ( K ) , A e [0, a ( K ' , x + ,u, u)]).

From Lemma 3.1 we can conclude that

n M1/~(K) = M ( K ) . nEN

Therefore it is sufficient to prove that M1/n(K) is a closed set for each n E N. The mapping

g~ " ~ A/'(K) x [0, l] --. Af(K) x

[ (x, u, t ) ~ (x, u, tcr(K ~, x + eu, u))

is continuous. Since A/'(K) × [0, l] is compact, the image set

H' (A f (K) x [0, 1]) = M~(K)

328 DANIEL HUG

is a compact subset of the topological Hausdorff space Af(K) × ~ and hence a closed set. []

The number n(K, p), p E K, has already been defined. It is not self-evident that the map p ~ n(K, p) is 7-/d-measurable on K. However, this statement is implicitly contained in Proposition 3.3.

PROPOSITION 3.3. Let K C IC d. Then we have

I(K) =/K n(K, y) dT-/d(y)

= /Ac [ ~ ( K , x , ~ ) ~ I1-- Ak,(K, x, u)l dT-/'()~) d '~d- l (x , U). (14) (K) J0 i----1 q l + kT(K , x, U) 2

Proof. Let us define W := Af(K) × [0, diam K] C I~ d × It( d × ~. From [9, Th. 3.2.23] we obtain that W is (7-/d, d)-rectifiable and trivially W is Borel measurable. We choose R > 0 sufficiently large, so that x - )~u E B d (o, R ) for (x, u ) E A/'(K) and )~ E [0, diam K]. The set Z := Bd(o, R) C ~d is (7-/d, d)-rectifiable and 7-/a-measurable. The situation described in [9, Th. 3.2.22] concerns a transforming map f , which is in our case given by

Af(K) > [0, diam K] ~ Z f .

( x , u , z -

and a ~ d L W-integrable function g chosen as

g : W -+ ~, g := 1M(K).

The characteristic function 1M(K)(X) is equal to 1 for x E M(K) and zero otherwise. It is also sufficient to assume that g is 7-/d L W-measurable and non- negative. An application of Federer's general area/coarea formula [9, Th. 3.2.22] yields

i ~ dT-/0 dT-gd(z). /w lM(K) ap Jdf dT-[d = i~ ff_i({z)) lM(K)

Since we explained in Section 2 that for 7-/d-almost all (x, u, )~) E W the relation

d-1 I1 - Aki(K, x, u)] ap Jdf(x, u, A) = i=,I-~ ~/1 + ki(K, x, u) 2


holds and because of

ff_,({.}) 1M(K)(X , U, A)dg°(x, u, A)= { n(K,o z) i f z E K

otherwise

(this confirms our measurability statement), we arrive at equation (14) after just another Fubini-like application of Federer's coarea theorem, i.e. [9, Th. 3.2.23]. []

For a convex body L E/Co d with o E int L, the radial function p (L, .) : II~ d\ { o} --+ it~ is defined by

p(L, x) := max{A _> 0lAx E L}, x E ~d\{o}.

Our next lemma extends and simplifies an argument in [13, (3.1.7) and pp. 45-46].

LEMMA 3.4. Let K, L E/Cg be convex bodies and o E int L. Then

Z d r q- 1 p ( i , dSd- l - r ( I~ , U) r----O

< V(K + L) - V(K). (15)

Equality holds, e.g., if K = Cd and L = DCd. Proof First, let K be a polytope. We use JZr(K ) to denote the system of r-

dimensional faces of K, r E {0, . . . , d - 1}. The relative interior of F E 5t'r(K) is denoted by relint F. Then the following disjoint partitioning of (K + L)\int K holds

d-1 (K + L ) \ i n t K = LJ LJ [ ( K + L)Mp(K, .)- '(relintF)].

~=o Fe:r~(K)

Further on, for r E {0, . . . , d - 1} and F E )Vr(K), we have the inclusion

relint F + {tu]t E [0, p(L, u)], u E N(K, F) NS d-l}

C (K + L) M p(K, -)-l(relint F). (16)

Notice that N(K, F) := N(K, x), x E relint F, is independent of the particular choice of x, see [20, p. 72]. Therefore we obtain from Fubini's theorem

330 DANIEL HUG

<_ ~d((K + L) N p(K, . )- l(relint F)) . (17)

Equality holds in (16) respectively (17) for K = Cd and L = DCd. If we use

E 7-[r(F)7-ld-I-r(N(K, F ) N "),

see [20, (4.2.11) and (4.2.18)], we get for r E {0 , . . . , d - 1}

<_ y~ Id((K + L) Np(K, . ) - l ( re l in tF) ) . Fe.%.(K)

This yields the lemma for a polytope K. The general case now is an immediate consequence of the weak continuity of the surface area measures. []

Proof of Theorem 1.1. The convexity of K implies for ~d- l -a lmost all (x, u) E Af(K) that ki(K, x, u) E [0, oo], i C {1, . . . , d - 1}. In the sequel we shall use the abbreviation

Hr(kl, . . . , kd-1) := ~ ~'~1''" ~lr, l<il<'"<ir<d-1

where k l , . . . , kd-1 E [0, oo] and r E {1 , . . . , d - 1}. In the special case r = 0 we define

H0(kl, . . . , kd-1):= 1.

We even write Hr(ki) , if it is clear from the context that i runs through the index set i = 1 , . . . , d - 1. Similar to [13] we introduce a rough estimate in (14):

S(K) dT--/1 (/~)dT-/d-1 (x, u) - J A r ( K ) 1 1 1 . 1 0 i--1 q l + ki(K, x, u) 2

--E

× ) r dT_/l()~)dT_/d-i(x, u)


d-1 1 far H~(ki(K, x, u)) + 1 r=o 1-Ii=l + x,

xa(K, x, u) ~+I dTgd-l(x, u)

_<

d - l l (

r-~0

d - l l (

r = 0

d ) /ar a(K, x, u)r+l dOd_1_r(K, x, u ) r + 1 (K)

d ) / H p(DK, u)r+ldOd-l-r(K,x,u) • r + 1 (K)

Again, analogous to [13, (3.1.4)], we have used the elementary estimate or(K, x, u) <_ p(DK, u) for all (x, u) C Af(K). This latter estimate enables us to get rid of the dependence on x E bdK, so that according to (11) we may transform the integrals over the generalized normal bundle into integrals over the unit sphere with respect to Aleksandrov's surface area measures:

d - l l ( d ) i s p(DK'u)r+ldSd-I-r(K'u)" I (K)<-~-d r + l e-1 r----0

From our unified approach we get with the help of Lemma 3.4, i.e.

d-ll ( d ) ~ ~ d r + l d-lp(DK'u)r+ldSd-'-~(K'u) v=O

<_ V(K + D K ) - V(K), (18)

the desired implication

I(K) V(K + DK)- V(K) n(K)- V(K) <- V(K) (19)

for an arbitrary convex body K E It0 d. For centrally symmetric K we even obtain the sharp estimate (equality holds for a d-cube)

n(t() < 3 d - 1. [] (20)

Whereas Lemma 3.1 merely provides pointwise convergence, Lemma 3.5 is formulated in terms of uniform convergence. In the proof of Theorem 1.3 this will be essential for the transition from convex bodies of class C~ ° to convex bodies all of whose boundary points are e-smooth. For the notion of an e-smooth boundary point, see [20, Notes for §2.5, 7].

332 DANmLHUG

LEMMA 3.5. Let K C t~ d be a convex body, e > O, and let Km C IC~, m E N, be a sequence of convex bodies of class C~ which satisfy K ~ C Km C K ~ + Ba(o, elm) for each m C N. Then there is a sequence/3(m), m E N, with lim,~___,~/3(rn) = 0 and, for all x E bd Kin,

I~(K~, x, aKin(x))- a(K ~, p(K ~, x), aK,(p(K ~, x)))l g fl(m).

Proof. For x E N~\int K ' we define s(x) E bd K C by the relation

{s(x)} = bd K ~ fq (p(K ~, x) + ]~>°(-aK,(p(gC , x)))).

The mapping s I bd K e : bd K ~ --+ bd K ~, x ~ s(x), is continuous, since for all x E bd K e

[x + Ncr/c,(x)] N int K ' ¢ O.

Moreover, for each x C bd K ' we have

71" <

As the spherical Gauss map aK, is continuous on bd K ' as well, there is a constant /3 such that, for all x E bd K ~,

71" _ <

This immediately implies for all x C Nd\int K ' that

7r / ( a K , ( 8 ( x ) ) , -ag~(p (K ~, x))) _ / 3 < ~-. (21)

Let x C bd Kin. According to our assumption on K,~, i.e. K ~ C Km C K ~(1 + (1 / m ) ) we have

IIx - p(K, x)ll _<E (1 + 1 ~ a n d Bd(p( K, E) C Kin. \ 'll~ /

Let C(x, Bd(p(K, x), e)) be the minimal closed cone with apex x that contains the ball Bd(p(K, x), e), and let a (x , m) be the angle between the axis of this cone and any of its extremal rays. Then

£

sin a(x, m) = IIx - p(K, x)ll E //2

so that

7r m Z(alcm(x), aK¢(p(K ~, x))) < ~ -- a r c s i n - -

- m + l


This last argument is taken from [15]. Similar to 8(x) we define the boundary point t,~(x) E bd K,~, for x E bd Kin, by the relation

]I~ >0 o" { t ~ ( ~ ) } = bd K,~ n ix + ( - ~m(x)) ] .

As a simple consequence of the triangle inequality we see for x E bd K,~:

IIx - t~ (x) l l _ IIx - p( K~, x)ll + IIp(K ~, x ) - 4 x ) l l + II~(x) - t ,~(x)ll,

that is,

E IIx - t ~ ( x ) l l - IIp(K ~, x) - s(x)ll ~ - - + II~(x) - tm(x)l l ,

m

as well as

I lp(K c, x) - 4 x ) l l ~ IIx - s(x)ll _ IIx - t~ (x) l l + Iltm(x) - s(x)ll .

Together this yields

IIIp(K c, x) - 8 ( x ) l l - IIx - t~(x) l l l _ & + 118(x)- tm(X)ll. m

To estimate the difference II ~(x ) - t ~ ( x )11, x E bd Kin, let us denote by C ( x, m ) the cone

C(x, .~):= • + {~ul (u, -,,K.(p(K', x))) _> cos c~(m), ,~ _> O, u E 5'd-1},

where x E bd Km and m E N. If we choose ra _> m0, m0 sufficiently large, and x E bd Kin, then all extremal rays of the cone C(x, m) hit the ball Bd(s(x) - eaK.(S(x)), e), according to (21) and limm--+~ a(rn) = 0. The set C(x, m)\Bd(s(x) - eaK~(S(x)), e), m >_ mo, is the union of two connected components. We denote the unbounded component by C+(x, m). Obviously we have

s(x), tin(x) E C+(x , m)N (K'(I+(1/'~))\K~) .

If we define

~ ( m ) :----- s u p d i a m [Ch(x, Tit)r'l (J~c(1-t-(1/ra))\l£e)]~ x~bd Km

we see that l i m m ~ 7(m) = 0. Therefore

E Illp( K~, ~) - ~ ( ~ ) l l - tl~ - t~(~) l l l <_ - + -~(m) =: Z(m)

m

334 DANIEL HUG

is the desired estimate. []

Proof of Theorem 1.3. First, we choose a sequence of convex bodies Kin, ra E N, as described in Lemma 3.5. For these convex bodies a mapping-degree argument yields:

(1 - (-1)d)V(Km) d - 1

= E fb o'(Km, x, aK,~(x))r+lH~(ki(Km, x))dT"ld-l(x) ( -1 ) ~

r=O 9~ -~ 1 dKm

: ( _ , ) T e - 1

r=O r + 1 r dKm

×ty(gm, x, CrK,,~(x)) r+l dCd-l-r(Km, x). (22)

This was essentially stated in [13, (4.1.5)]. To get (22), we used the representation of the curvature measures in the C~ case [20, (4.2.19)1 as integrals of the elementary symmetric functions of the principal curvatures ki(Km, x) of bd I(m. We set

It(m) : : fb ~r(Km, x, aKin(X)) r+l dCd-l-r(Km, x). dKm

The mapping

is defined on ]Rd\int K ~ and continuous, since

(p(K ~, x) + I~aK,(p(K e, x))) nint K ~ ¢ 0.

With the help of Lemma 3.5, the following estimate is justified

/ r ( m ) - fbdKm cr(K e, p(I(e,x), CrK,(p(K e, x))) r+l dCd-l-r(Km, x)[

-< fb const-/3(m) dCa-l-~(K,~, x) dKm

= const-/3(m)Cd_l_r(K,~, lRd).


The last expression tends to zero for m ~ c~, as K,~ --+ K ~ implies Cd-I-T(Km, it, d) ~ Cd_I_,.(K ~, gd). Because of the weak continuity of Federer's curvature measures, we observe that

lim f a(K c, p(K ~, x), aif~(p(K ~, x))) r+l dCd-l-,(If,~, x) m--+oo Jbd Km

= fbdga(Ke , p (g ~, x), aK,(p(K ~, x))) ~+1 dCd_l_r(I( ~, x)

-~- fbdK" 6r(g~' x, aKe(X)) r+l dCd_l_r(g c, x).

Hence, we arrive at

(1 - ( - 1 ) g Y ( g 9

( ) = ~--~ ( -1 ) ~ d - 1

~=0 r + l r

fbdK a(g¢' aK,(x)) ~+1 dCd_l_~(I( ¢, x). (23) X ,~ X~

Obviously lim40 V (K ~) = V(K). Therefore we have to show that the fight side of (23) converges to the fight side of (4) for e + 0. To this end the integrals on the right side of (23) will first be transformed into integrals over Af(K) by repeatedly using Federer's coarea formula:

( d - 1 ) f b r dK ~ cr(I(C' X' aK'(X))r+! dCd-l-r(KC' x)

(12) ~bdh" a ( g ~, x, aK,(x))r+lHr(ki(g ~, x, ag~(X)))dT-[d-l(x)

(7) f Z ap Jd-lt,(x, u)a(K', x + eu, u ) r + l (g)

×H~(ki(g c, x + eu, u)) dT-ld-l(x, u).

Finally, an application of Lebesgue's bounded convergence theorem [9, §2.4.9] shows that it is sufficient to prove that

ap Jd-lt~(x, u)a(K ~, x + eu, u)~+lH~(ki(K~, x + eu, u)) (24)

is bounded almost everywhere on Af(K) by a constant independent of e, and further on that this expression converges almost everywhere on Af(K) to

or(I(, x, u) ~+lHr(k l (g ' x, u) , . . . , kd-l(g, x, u)) (25) d-1 ~/1 ki(K, u) 2 1-Ii=l + x,

336 DANIEL HUG

for e ~ 0. Boundedness may be seen from a(K ~, x + cu, u) <_ diam K e and from the subsequent estimates, where we assume without loss of generality that 0 < e < 1 andkl(K, x, u) >_ .. . >_ kd-l(K, x, u) >_ O:

ap Jd-l tdx, ~)nr(k~(K', • + ~ , ~))

l +ek,(K, x, u) ( d - 1 ) f l kj(K, x, u) ¢1 + ki(K, x, u) 2 r 1 + ekj(K, x, u) 4=1 j=l

( d - l ~ ~- T ki(K, x, u) d-1 1+ ¢ki(K, x, u)

i=1 ~/1 + ki(K, x, "= 1 ¢1 + ki(K, x, U) 2

Besides we notice, if we write ki := ki( K, x, u)andki(K ~) := ki( K ~, x + eu, u) for the moment and { j l , . . . , j e - I -T} , 1 _< j l < "'" < jd-l-T _< d - 1, for the index set complementary to { i l , . . . , iT}, 1 _< il < . . . < iv <_ d - 1 , with respect t o { l , . . . , d-l}"

d-1 1-I m+eki Z i=| ~-1- k2 l_~'/l<'"<~r~_d-1

kil ( I(~) . . . kir ( I(C)

{( ~lTekil } = l<i,<...~</r_<d-1 \i=l ~ ] ~il(t£¢)"'kir(K¢)

{ ( ~ ! " J l - E k l I k~l ~ir } (:-) Z ~ ] 1+ ,k,, "'" : + ~k,~

l<_il<'"<ir<d-1 \i=1

= kil d - l - r l + ¢ k j t

I=1 l - i2l H " + t=l

Now we have

1 + ~kjt 1 --'+ ¢1 + k 2. ¢1 + k 2. 3t 3t

fore ~ O.

This is shown by distinguishing the two cases kit = co and kit < oc. These considerations are all valid for 7-/d-l-almost every (x, u) ~ Af(K). The situation here is essentially the same as the one discussed in [15, p. 35]. Lemma 3.1 then

NORMALS THROUGH A POINT IN A CONVEX BODY

completes the proof for the convergence of (24) to (25).

337

[]

4. Miscellanea

There seems to be little hope to obtain a complete description of those cases in which equality occurs in the right inequality of (2). However, in the plane and especially for polygons we can draw some conclusions from the above investigations which refine statements of [13].

For an equilateral triangle To E/C~ it is shown in [13] that n(To) = 6. A more detailed statement can be found in the following proposition.

PROPOSITION 4.1. For an arbitrary triangle T E IC 2 the inequalities

4 < n(T)_< 6

hold. For a triangle T the equality n( T ) = 6 holds if and only if the internal angles o f T do not exceed ½~r. In addition, for each r E (4, 6] there is a triangle T(r) with n(T(r)) : r.

We already know the general estimate n(K) _< 8 for centrally symmetric K E /C0 2. In the class of all centrally symmetric convex polygons K E /C0 2 we shall characterize those polygons for which the upper hound is attained.

PROPOSITION 4.2. Let K E IC 2 be a centrally symmetric polygon with edges F1, . . . , F2m in cyclic order. Then the following two statements are equivalent.

(a) n(K) = 8. (b) For each i E {1 , . . . , m} the convex hull conv{F/U F~+i } is a rectangle.

Proof of Propositions 4.1 and 4.2. We start with a derivation of the inequality I ( K ) <_ 8V(K) for K E /C 2, K = - K , along the lines of the proofs for Proposition 3.3 and Theorem 1.1:

±(K) = f fa(K,x,u) I i _ Ak(K, x, u)] dTyl(A)d~l(x ' u) aJC(K) go ¢1 + k(K, x, u) 2

/X /o'(K,x,u) 1 "Jv ~]¢(I(_', Z, it) d?_~l(/~)d.~l(x, u) - (K) gO ¢ 1 + k(K, x, u) 2

= / H a(K, x, u )dOl (K , x, u) (K)

1 L + 2 (K) a (K, x, u) 2 d®o(K, x, u)

3 3 8 DANmL rrtJG

f

= 2J2c(K)a(K, x, u)dOl (K , x, u)

<_ 2].]¢(K)P(DK, u) dOl(K, x, u)

1 Jfs h(K, u)dSI(K, u)= 8V(K).

,

Here, we used Theorem 1.3 for d = 2, p(DK, u ) = 2p(K, u) _< 2h(K, u), and a very special case of [20, (5.1.18)]. In ® equality occurs if and only if for ~l-almost all(x, u) C N'( K ) we either have k( K, x, u)=Oork(K, x, u ) = oo. Thus for a polygon equality holds in ®. It should be remarked that this is true for a typical convex body as we shall see in Theorem 4.5. Now, we observe that I(K) = 8V(K) holds if and only if

a(K, x, u)= 2p(K, u)= 2h(K, u)

is true for E)i(K, .)-almost all (x, u) C Af(K). For a polytope K as described in the assumptions of Proposition 4.2 this last condition may be paraphrased by saying that

a(K, x, u i ) = 2p(g , u i ) = 2h(g , ui) (26)

must hold true for i E { 1 , . . . , 2m} and for 7-/1-almost all (hence for all) x C F/, if ui is the exterior unit normal vector to F/. A geometric interpretation of (26) yields the statement of Proposition 4.2.

In the special case of a triangle T E/C~ the above reasoning yields

I(T) = 2 f~(T)a(T, x, u)d0t(T, x, u)

3

= 2 ~ [~ a(T, x, ui) d~[l(x) i = l

3

<_ 2 ~ V(T) = 6V(T), i = l

if we denote by Ft, F2, F3 the edges of T with corresponding exterior unit normal vectors ul, u2, u3. The inequality

fF a(T, ui) dT-/l(x) _< V(T)

results from Fubini's theorem and the meaning of the function a(T, .). The dis- cussion of the equality case is obvious now.


Let us consider an isosceles triangle T~ whose baselength is equal to 1 and the adjacent internal angles of which are equal to a E (0, 3]" An elementary geometric argument leads to

[ 2 ] I (T~) = 2 + cos2 c~

This proves the last statement of Proposition 4.1. []

In the proof of Proposition 4.2 we announced a result on the generic boundary behaviour of convex bodies. A far reaching result of this type was discovered by Zamfirescu [23, Th. 1]. As a matter of fact the proof of our next theorem is so similar to the one given in [23] or in [20, Th. 2.6.2] that we can omit it here. Nevertheless, our theorem cannot be regarded as a corollary to Zamfirescu's result, since e-smooth convex bodies merely form a meagre set in/Co d.

THEOREM 4.3. Let c > 0 be given. For most convex bodies K in lC~, at every boundary point y C bd K ~, and for every tangent direction t to bd K ~ at y,

p~(K ~, y, t) = e or p , ( K e, y, t) = oo.

For a definition of these lower and upper radii of curvature see, e.g., [20, §2.5].

COROLLARY 4.4. Let e > 0 be given. For most convex bodies K in Edo the normal curvature o f K e at y in direction t is

1 ~ ( K ~, y, t ) = - or ~ ( K ~, y, t ) = o

E

for 7-[ d- 1 -almost all y C bd K e and every tangent direction t at y. Proof. This follows immediately from Aleksandrov's [2] theorem on the almost

everywhere existence of second differentials of a convex function together with Theorem 4.3. []

Finally, the definition of the generalized curvatures ki(K, x, u), i E {1 , . . . , d - 1 }, leads to our next theorem.

THEOREM 4.5. For most convex bodies K in I~ , for 7-[ d-1 -almost all (x, u) C Af(K) , and fora l l i E {1 , . . . , d - 1},

k (K, x , u) = 0 or k (K, x, u) =

COROLLARY 4.6. For most convex bodies K in IC d

I ( K ) = E d r + 1 x, u) r+l d O d - r - l ( K , x, g). r=O

340 DANIEL HUG

Acknowledgement

The author wishes to thank Professor Rolf Schneider for his encouragement and support.

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meinerung einiger Begriffe der Theorie der konvexen KOrper (in Russian), Mat. Sbornik N.S. 2 (1937), 947-972.

2. Aleksandrov, A. D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it (in Russian), Uchenye Zapiski Leningrad. Gos. Univ., Math. Ser. 6 (1939), 3-35.

3. Bangert, V.: Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannig- faltigkeiten, Z reine angew. Math. 307/308 (1979), 309-324.

4. Blaschke, W.: Einige Bemerkungen tiber Kurven und Flachen yon konstanter Breite, Ber. Verh. Si~chs. Akad. Leipzig 67 (1915), 290-297.

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10. Fenchel, W. and Jessen, B.: Mengenfunktionen und konvexe KOrper, Danske Vid. Selskab Mat.- fys. Medd. 16 (3) (1938), 1-31.

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13. Hann, K.: The average number of normals through a point in a convex body and a related Euler-type identity, Geom. Dedicata 48 (1993), 27-55.

14. Hell, E.: Concurrent normals and critical points under weak smoothness assumptions, Ann. New YorkAcad. Sci. 440 (1985), 170-178.

15. Kohlmann, P.: KrtimmungsmaBe, Minkowskigleichungen und Charakterisierungen metrischer B~ille in Raumformen, Dissertation, Universit~t Dortmund, 1988.

16. Kohlmann, P.: Curvature measures and Steiner formulae in space forms, Geom. Dedicata 40 (1991), 191-211.

17. Rogers, C. A.: HausdorffMeasures, Cambridge University Press, New York, 1970. 18. Santal6, L. A.: Note on convex spherical curves, Bull. Amer. Math. Soc. 50 (1944), 528-534. 19. Santal6, L. A.: Sobre los cuerpos convexos de anchura constante in E,~, PortugalMath. 5 (1946),

195-201. 20. Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory, Encycl. of Math. and its Appl.,

vol. 44, Cambridge University Press, Cambridge, 1993. 21. Walter, R.: Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ.

Hamburg 45 (1976), 263-282. 22. Z~ihle, M.: Integral and Current representation of Federer's curvature measures, Arch. Math. 46

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