Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 185-193.

On the Affine Rectification of Convex Curves

Kurt Leichtweiß

Mathematisches Institut B, Universität Stuttgart Pfaffenwaldring 57, D-70550 Stuttgart

1. Introduction

In the following let C be a plane convex curve, i.e. a connected, nonoriented part of the boundary of an arbitrary twodimensional convex region K in the affine plane 𝔸^{n} with the support function h(u) (u ∈ S^{1}). The curve

C : x = x(t) (α ≦ t ≦ β)  (1)

may be either closed or an arc of bd K. We attach to C the set 𝒯(C) of all of its support elements, consisting of a point x(t) together with a support line H_{h(u)}(u) of K, passing through x(t) (u_{−}(t) ≦ u ≦ u_{+}(t)) ^{1} , with the outer unit normal vector u (at the different endpoints x(α) and x(β) of an arc C final support elements with the support lines H_{h(u_{−})}(u_{−}) (u_{−} ≦ u_{+}(α)) and H_{h(u_{+})}(u_{+}) (u_{−}(β) ≦ u_{+}) may be defined).

It was the idea of G. Pick and W. Blaschke to inscribe into this generalized tangent bundle 𝒯(C) of C the union ⋃_{l=1}^{k} 𝒫_{l−1l} of parabola arcs 𝒫_{l−1l}, determined by their final support elements ℒ_{l−1} := (x(t_{l−1}), H_{h(u_{l−1})}(u_{l−1})) and ℒ_{l} := (x(t_{l}), H_{h(u_{l})}(u_{l})) belonging to a certain subdivision S of 𝒯(C) (l = 1, ..., k; t_{0} = α, u_{0} ≦ u_{+}(α), t_{k} = β, u_{k} ≧ u_{−}(β)). This is possible if we suppose 0 < ∡(u_{l−1}, u_{l}) < π and exclude the degenerate cases x(t_{l−1}) ∈ H_{h(u_{l})}(u_{l}) or x(t_{l}) ∈ H_{h(u_{l−1})}(u_{l−1}) where we have to replace the parabola arc by a line segment of C. The used method is the equiaffine analogue to the euclidean method of inscribing a polygon because a parabola arc has vanishing affine curvature in analogy to the fact that a line segment has vanishing euclidean curvature. Since the equiaffine arclength of 𝒫_{l−1l} (also in the degenerate case) equals 2V (Δ_{l−1l})^{\frac{1}{3}} where V (Δ_{l−1l}) is the twodimensional volume of the triangle Δ_{l−1l} with the vertices x(t_{l−1}), x(t_{l}) and H_{h(u_{l−1})}(u_{l−1}) ∩ H_{h(u_{l})}(u_{l}) (see [1], § 3) it makes sense to define

d^{a}(ℒ_{l−1}, ℒ_{l}) := 2 · V (Δ_{l−1l})^{\frac{1}{3}}  (2)

1We identify the points u of S^{1} with their polar angles.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

as the affine distance of the final support elements ℒ_{l−1} and ℒ_{l} of 𝒫_{l−1l}. The so defined affine distance fulfils the (inverse) affine triangle inequality

d^{a}(ℒ_{1}, ℒ_{2}) ≧ d^{a}(ℒ_{1}, ℒ_{3}) + d^{a}(ℒ_{3}, ℒ_{2})  (3)

if ℒ_{3} is an intermediate support element of the support elements ℒ_{1} and ℒ_{2} of 𝒯(C). Here equality holds precisely in the trivial cases t_{1} = t_{2}, u_{1} < u_{2} or t_{1} < t_{2}, u_{1} = u_{2} or t_{1} < t_{2}, u_{1} = u_{12} < u_{2} resp. u_{1} < u_{12} = u_{2} where all affine distances vanish and in the nontrivial case t_{1} < t_{2}, u_{1} < u_{12} < u_{2} for any support element ℒ_{3} of the parabola arc 𝒫 with the final support elements ℒ_{1} and ℒ_{2} (see [1], § 16; u_{12} is the outer unit normal vector of the segment from x(t_{1}) to x(t_{2})).

For all these reasons we introduce an equiaffinely invariant arclength L(𝒯(C)) for the generalized tangent bundle 𝒯(C) of C by

Definition 1.1. The (always existing) nonnegative entity

L(𝒯(C)) := inf_{S} (∑_{l=1}^{k} d^{a}(ℒ_{l−1}, ℒ_{l})) = inf_{S} (∑_{l=1}^{k} 2 · V (Δ_{l−1l})^{\frac{1}{3}} )  (4)

is called affine arclength of 𝒯(C) ^{2} (affine rectification).

From Definition 1.1 we may derive

Proposition 1.2. a) If C_{1} ∪ C_{2} is a simple convex curve (closed or an arc) then

L(𝒯(C_{1} ∪ C_{2})) = L(𝒯(C_{1})) + L(𝒯(C_{2})).  (5)
b) If ℒ_{1} and ℒ_{2} are the final support elements of 𝒯(C) (∡(u_{1}, u_{2}) < π) then
L(𝒯(C)) ≦ d^{a}(ℒ_{1}, ℒ_{2})  (6)

with equality only for d^{a}(ℒ_{1}, ℒ_{2}) = 0 or if C equals the parabola arc 𝒫 with 𝒯(𝒫) connecting ℒ_{1} and ℒ_{2}.

Proof. a) The additivity property (5) is a consequence of the fact that 𝒯(C_{1} ∪ C_{2}) = 𝒯(C_{1}) ∪ 𝒯(C_{2}) with 𝒯(C_{1})∩𝒯(C_{2}) =: ℒ_{3} for suitable chosen 𝒯(C_{1}) and 𝒯(C_{2}) at the end of C_{1} and the beginning of C_{2} and of Definition 1.1 since because of (3) there does not exist any restriction for the infimum in (4) (with C = C_{1} ∪ C_{2}) if each subdivision of 𝒯(C_{1} ∪ C_{2}) contains ℒ_{3}.

b) The inequality (6) is an immediate consequence of (3) and (4). Herein we deduce from the equality in

L(𝒯(C)) = L(𝒯(C_{1})) + L(𝒯(C_{2})) ≦ d^{a}(ℒ_{1}, ℒ_{3}) + d^{a}(ℒ_{3}, ℒ_{2}) ≦ d^{a}(ℒ_{1}, ℒ_{2})  (7)

2Here it must be supposed that d^{a}(ℒ_{l−1}, ℒ_{l}) is defined, i.e. 0 ≦ ∡(u_{l−1}, u_{l}) < π.

(see (5), (6) and (3)) that either d^{a}(ℒ_{1}, ℒ_{2}) = 0 or that the (arbitrarily chosen) intermediate support element ℒ_{3} of C lies in the generalized tangent bundle 𝒯(𝒫). This completes the proof of Proposition 1.2. □

2. Onedimensional affine surface area for an arc of the boundary of a twodimensional plane convex region

Now we start with an arc C of bd K which also may be closed and state that it has been defined for C the notion of a (onedimensional) affine surface area A(C) of C by the Lebesgue integral

A(C) := ∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) = lim_{δ→+0} c_{n} · \frac{V (K, C, δ)}{δ^{\frac{2}{n+1}}}  (8)

(C. Schütt and E. Werner [8]). Here κ is the euclidean curvature existing at the normal points of C, i.e. almost everywhere (see [7], theorem 2.5.5), and ℋ^{1} is the onedimensional Hausdorff measure on C. Further c_{n} is a dimension depending constant and V (K, C, δ) the volume of the shell between K_{δ} and K over the arc C of bd K where K_{δ} is the convex floating body of K to the parameter δ > 0 (see [8]). According to the transformation formula

∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) = ∫_{u(C)} Δ(h)(u)^{\frac{2}{3}} dω(u)  (9)

of D. Hug (see [3], section 2) for the spherical map the affine surface area may also be expressed by

A(C) = ∫_{u(C)} Δ(h)(u)^{\frac{2}{3}} dω(u)  (10)

(Leichtweiß [4]) with the twodimensional Laplace operator Δ and the Lebesgue measure ω(·) on S^{1}. Because of the equivalence of the representation (10) of A(bd K) = A(K) with a representation of A(K) by E. Lutwak as an infimum (see [5] and [2]) it follows that the functional A is upper semicontinuous in the Hausdorff sense, i.e.

lim sup_{ν→∞} A(K_{ν}) ≦ A(K)  (11)

for every sequence {K_{ν}}_{ν∈ℕ} of plane convex regions converging to K.

3. Main result

In the classical case of a convex curve C of differentiability class 𝒞_{+}^{3} W. Blaschke (see [1], § 4) proved the coincidence of the result (4) of the affine rectification of C with the onedimensional surface area A(C) of C (see (8). The main result of this paper says that this remains to be true if the smoothness assumption for C is omitted:

Theorem 3.1. For every arc C of the boundary of a twodimensional convex region K in 𝔸^{2} we have the equality

inf_{S} (∑_{l=1}^{k} d^{a}(ℒ_{l−1}, ℒ_{l})) = ∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x). ^{3} (12)

4. Proof of the main result

We shall prove Theorem 3.1 under the stronger assumption that the final support elements ℒ_{0} and ℒ_{k} of the arc

C : x = x(t) (α ≦ t ≦ β)  (13)

have outer unit normal vectors u_{−} and u_{+} with

0 ≦ ∡(u_{−}, u_{+}) < π.  (14)

Here (14) means no loss of generality as we can get this relation by subdividing C and as both parts of the equality (12) have the additivity property (see (5)).

Now, in a first step of the proof we will show the inequality

∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) ≦ inf_{S} (∑_{l=1}^{k} d^{a}(ℒ_{l−1}, ℒ_{l})) =: L(𝒯(C)).  (15)

After the consideration just made above it suffices to prove

∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) ≦ d^{a}(ℒ_{0}, ℒ_{k}) = 2 · V (Δ_{0k})^{\frac{1}{3}}  (16)

where V (Δ_{0k}) is the area of the triangle with the vertices p_{0} := x(α), p_{k} := x(β) and p_{0k} := H_{h(u_{−})}(u_{−})∩H_{h(u_{+})}(u_{+}) if ℒ_{0} := (x(α), H_{h(u_{−})}(u_{−})) and ℒ_{k} := (x(β), H_{h(u_{+})}(u_{+})) are the final support elements of C ^{4}. The inequality (16) is trivial if p_{0} ∈ H_{h(u_{+})}(u_{+}) or p_{k} ∈ H_{h(u_{−})}(u_{−}) and therefore V (Δ_{0k}) = 0 because then C is a line segment with ∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) = 0. For this reason we may additionally suppose

p_{0} ∉ H_{h(u_{+})}(u_{+}), p_{k} ∉ H_{h(u_{−})}(u_{−})  (17)

and thus

∡(u_{−}, u_{+}) > 0.  (18)

Now the proof of (16) will be made in an indirect manner. We assume

∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) > 2 · V (Δ_{0k})^{\frac{1}{3}}  (19)

3Despite the euclidean constructions this result is purely equiaffinely invariant.

4In the case u_{−} = u_{+} we have to read V (Δ_{0k}) = 0.

and come to a contradiction. At first we approximate the arc C : x = x(t) (t_{0} ≦ t ≦ t_{k}) with the support elements (x(t), H_{h(u)}(u)), where u_{−}(t) ≦ u ≦ u_{+}(t) and especially u_{−} ≦ u_{+}(t_{0}), u_{−}(t_{k}) ≦ u_{+}, by the outer parallel arc C^{ρ} with the support elements (x(t) + ρu, H_{h(u)+ρ}(u)) (ρ > 0). Then the arc C^{ρ} of differentiability class 𝒞^{1} has ρ(x)-smooth points with ρ(x) ≧ ρ ^{5}, and its affine surface area has the value

A(C^{ρ}) := ∫_{C^{ρ}}(κ^{ρ})(x)^{\frac{1}{3}} d(ℋ^{ρ})^{1}(x) = ∫_{u_{−}}^{u_{+}} Δ(h^{ρ})(u)^{\frac{2}{3}} dω(u)

= ∫_{u_{−}}^{u_{+}}(Δh + ρ · 1)(u)^{\frac{2}{3}} dω(u) (ρ ≧ 0)  (20)

(see (9)). But (20) implies

lim_{ρ→+0} A(C^{ρ}) = A(C)  (21)

by Levi’s theorem on the monotone convergence such that ρ may be chosen so small that, by a continuity argument, (19) and (17) also hold for C^{ρ} instead of C. Therefore we go further assuming that (19) as well as (17) and (18) are valid for an arc C of class 𝒞^{1} with ρ(x)-smooth points where ρ(x) ≧ ρ > 0.

In this case the spherical map σ^{(C)} of C is well-defined and Lipschitzian (see [4], Hilfssatz 1). If we introduce the euclidean arclength parameter σ on C by

σ(t) := ∫_{x(t_{0})}^{x(t)} dℋ^{1}(x) = ∫_{t_{0}}^{t} \sqrt[+]{(\frac{dξ_{1}}{dt} )^{2} (τ) + (\frac{dξ_{2}}{dt} )^{2} (τ)} dτ  (22)

(x = (ξ_{1}, ξ_{2})) and the euclidean arclength parameter Θ on the circle S^{1} by

Θ(u) := arctan \frac{υ_{2}}{υ_{1}}  (23)

(u = (υ_{1}, υ_{2})) the Lipschitz property of σ^{(C)} implies its absolute continuity and therefore

Θ(σ) = ∫_{0}^{σ} \frac{dΘ}{dσ}(ς) dς + Θ(0).  (24)

Here the derivative \frac{dΘ}{dσ} exists almost everywhere, represents a measurable function and moreover we have by [3], lemma 2.3, for the almost everywhere on C existing normal points x(σ) the relation

\frac{dΘ}{dσ}(σ) = κ(σ).  (25)

In the same time the Lipschitz property of σ^{(C)} obviously yields

|κ(σ)| = |\frac{dΘ}{dσ}(σ)| ≦ M.  (26)

5A boundary point x of a convex region K is called ρ(x)-smooth if ρ(x) ≧ 0 is the maximal radius of a ball B_{ρ(x)} with x ∈ B_{ρ(x)} ⫅ K.

By this way the function κ(σ) determines the arc C in a unique manner by

ξ_{1}(σ) = ∫_{0}^{σ} cos (Θ(ς) + \frac{π}{2} ) dς + ξ_{1}(0),

ξ_{2}(σ) = ∫_{0}^{σ} sin (Θ(ς) + \frac{π}{2} ) dς + ξ_{2}(0)  (27)

where Θ(ς) is given by (24) and (25).

It is now our aim to approximate C by a 𝒞^{∞}-smooth convex arc C̃ which also has the properties (17) and (19) of C. For this purpose we consider the function κ(σ)^{\frac{1}{3}} . It is measurable like κ(σ) and bounded:

0 ≦ κ(σ)^{\frac{1}{3}} ≦ M^{\frac{1}{3}}  (28)

(see (26)). Hence it is an element of the function space L^{2}[0, ℓ] (ℓ := σ(t_{k}) equals the euclidean arclength of C). Now we apply the well-known fact that κ(σ)^{\frac{1}{3}} may be approximated in the quadratic mean by a Fourier polynomial p(σ) on the interval [0, ℓ] with prescribed accuracy δ > 0, i.e.

∥κ^{\frac{1}{3}} − p∥_{L^{2}[0,ℓ]} := (∫_{0}^{ℓ}(κ(σ)^{\frac{1}{3}} − p(σ))^{2} dσ)^{\frac{1}{2}} < δ  (29)

(see [6], p. 60–62). This inequality remains valid if p(σ) will be truncated from below and above by the constant functions 0 and M^{\frac{1}{3}} followed by a suitable smoothing procedure (see (28)). By this way we obtain a 𝒞^{∞}-smooth function p̃ on [0, ℓ] with the property

∫_{0}^{ℓ}(κ(σ)^{\frac{1}{3}} − p̃(σ))^{2} dσ < δ^{2} where 0 ≦ p̃(σ) ≦ M^{\frac{1}{3}} .  (30)

Now we define the 𝒞^{∞}-smooth function Θ̃(σ) on [0, ℓ] by

Θ̃(σ) = ∫_{0}^{σ} p̃(ς)^{3} dς + Θ(0)  (31)

which determines the 𝒞^{∞}-smooth convex arc C̃, given by

ξ̃_{1}(σ) = ∫_{0}^{σ} cos (Θ̃(ς) + \frac{π}{2} ) dς + ξ_{1}(0),

ξ̃_{2}(σ) = ∫_{0}^{σ} sin (Θ̃(ς) + \frac{π}{2} ) dς + ξ_{2}(0).  (32)

Clearly σ with σ ∈ [0, ℓ] is also the euclidean arclength parameter for C̃ for which ũ(σ) := (cos Θ̃(σ), sin Θ̃(σ)) is the unit normal vector and p̃(σ)^{3} ≧ 0 the curvature at the point (ξ̃_{1}(σ), ξ̃_{2}(σ)). Therefore the arcs C and C̃ are touching at their starting points with σ = 0. They do not differ too much because of the estimation

|Θ̃(σ) − Θ(σ)| = |∫_{0}^{σ}(p̃(ς)^{3} − κ(ς)) dς|

= |∫_{0}^{σ}(p̃(ς) − κ(ς)^{\frac{1}{3}} ) · (p̃(ς)^{2} + p̃(ς)κ(ς)^{\frac{1}{3}} + κ(ς)^{\frac{2}{3}} ) dς|

≦ (∫_{0}^{σ}(p̃(ς) − κ(ς)^{\frac{1}{3}} )^{2} dς)^{\frac{1}{2}} · 3M^{\frac{2}{3}} σ^{\frac{1}{2}} ≦ 3M^{\frac{2}{3}} δ · σ^{\frac{1}{2}} (0 ≦ σ ≦ ℓ)  (33)

(see (31), (24), (25), (28) and (30)) which implies now in the same manner because of (32) and the corresponding relations for the original arc C of class 𝒞^{1} that for i = 1, 2

|ξ̃_{i}(σ) − ξ_{i}(σ)| ≦ ∫_{0}^{σ} |Θ̃(ς) − Θ(ς)| dς ≦ 2M^{\frac{2}{3}} δ · σ^{\frac{3}{2}} (0 ≦ σ ≦ ℓ).  (34)

Here we have used the Lipschitz property of the functions cos and sin with the Lipschitz constant 1. Since we also have for the affine surface areas A(C̃) and A(C) the relation

|A(C̃) − A(C)| = |∫_{0}^{ℓ}(p̃(σ) − κ(σ)^{\frac{1}{3}} ) dσ|

≦ (∫_{0}^{ℓ}(p̃(σ) − κ(σ)^{\frac{1}{3}} )^{2} dσ)^{\frac{1}{2}} · (∫_{0}^{ℓ} dσ)^{\frac{1}{2}} < δ · ℓ^{\frac{1}{2}}  (35)

(see (30)) we see by (35) as well as by (33) and (34) for σ = ℓ that δ may be chosen so small that by a continuity argument (19) and (17) also hold for C̃ instead of C.

Now we may proceed assuming that (19) as well as (17) and (18) are valid for an arc C of class 𝒞^{∞}. For a further computation it is convenient to introduce affine coordinates {ξ_{1}, ξ_{2}} in the plane 𝔸^{2} requiring that the affinely independent points p_{0}, p_{k} and p_{0k} (see (17) and (18) ^{6}) have the coordinates

p_{0k} = (0, 0), p_{0} = (−1, 1) and p_{k} = (1, 1).  (36)

Then C may be represented by

ξ_{1}(t) = t, ξ_{2}(t) = f(t) (−1 ≦ t ≦ 1)  (37)

(f of class 𝒞^{∞}) because of its convexity and the fact that the lines ξ_{1} + ξ_{2} = 0 and ξ_{1} − ξ_{2} = 0 are the tangential lines of C at its ends t = −1 and t = 1. This implies for the derivative ḟ of f with respect to t the relations

ḟ(−1) = −1, ḟ(1) = 1.  (38)

6Without loss of generality we have here additionally assumed the validity of (14) by a continuity argument.

Hence we obtain after application of the Hölder inequality

∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x) = A(C) = ∫_{−1}^{1} | \begin{matrix}1 & ḟ(t) \\ 0 & f̈(t) \end{matrix} |^{\frac{1}{3}} dt = ∫_{−1}^{1} f̈(t)^{\frac{1}{3}} dt

≦ (∫_{−1}^{1} f̈(t)dt)^{\frac{1}{3}} · (∫_{−1}^{1} dt)^{\frac{2}{3}} = (ḟ(1) − ḟ(−1))^{\frac{1}{3}} · 2^{\frac{2}{3}} = 2 = 2 · V (Δ_{0k})^{\frac{1}{3}}  (39)

as obviously V (Δ_{0k})^{\frac{1}{3}} equals 1. But (39) contradicts the assumption (19) for C and this contradiction shows that (16) and therefore also (15) are valid.

Now in a second step we will prove the inverse inequality

L(𝒯(C)) ≦ ∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x)  (40)

for every arc C : x = x(t) (t_{0} ≦ t ≦ t_{k}) of a closed convex curve C ∪ C′ in 𝔸^{2} where 𝒯(C) ∩ 𝒯(C′) is the union of the common final support elements of C and C′. Here it is

the idea to interpret ∑_{l=1}^{k} d^{a}(ℒ_{l−1}, ℒ_{l}) = ∑_{l=1}^{k} 2 · V (Δ_{l−1l})^{\frac{1}{3}} as affine surface area A(Ĉ) of an arc Ĉ which results from C if we replace the subarcs C_{l−1l} of C from x(t_{l−1}) to x(t_{l}) in the nondegenerate case x(t_{l−1}) ∉ H_{h(u_{l})}(u_{l}) and x(t_{l}) ∉ H_{h(u_{l−1})}(u_{l−1}) and thus ∡(u_{l−1}, u_{l}) > 0 (compare (17)) by the parabola arcs 𝒫_{l−1l} determined by their final support elements ℒ_{l−1} = (x(t_{l−1}), H_{h(u_{l−1})}(u_{l−1})) and ℒ_{l} = (x(t_{l}), H_{h(u_{l})}(u_{l})) (l = 1, ..., k). This is possible since we have equality in the inequality (39) for 𝒫_{l−1l} with the representation

ξ_{1}(t) = t, ξ_{2}(t) = \frac{1}{2}(t^{2} + 1) (−1 ≦ t ≦ 1) ^{7}  (41)

and since we have A(C_{l−1l}) = 0 = 2 · V (Δ_{l−1l})^{\frac{1}{3}} in the degenerate case.

Now we start with the definition (4) of L(𝒯(C)). As an infimum L(𝒯(C)) may also be represented as a limit in the following manner:

L(𝒯(C)) = lim_{ν→∞} (∑_{l_{ν}=1}^{k_{ν}} d^{a}(ℒ_{l_{ν}−1}, ℒ_{l_{ν}})) = lim_{ν→∞} A(Ĉ_{ν}).  (42)

By a suitable subdividing procedure for the subarcs C_{l_{ν}−1l_{ν}} which does not cancel the infimum nature of the limit (42) because of (3) we achieve without loss of generality the additional property

tan ∡(u_{l_{ν}−1}, u_{l_{ν}}) < \frac{1}{ν} (l_{ν} = 1, ..., k_{ν})  (43)

for all ν ∈ ℕ. Now the absolute value of the difference of the support functions h resp. ĥ_{ν} of C resp. Ĉ_{ν} on S^{1} along the subarc C_{l_{ν}−1l_{ν}} may be estimated from above by the height Δ_{l_{ν}} of the triangle Δ_{l_{ν}−1l_{ν}} over its side [p_{l_{ν}−1}, p_{l_{ν}}] because C_{l_{ν}−1l_{ν}} and 𝒫_{l_{ν}−1l_{ν}} are contained in Δ_{l_{ν}−1l_{ν}}:

7Without loss of generality we have here assumed ∡(u_{l−1}, u_{l}) < π (compare footnote 6).

|ĥ_{ν}(u) − h(u)| ≦ Δ_{l_{ν}} (u_{l_{ν}−1} ≦ u ≦ u_{l_{ν}}; 1 ≦ l_{ν} ≦ k_{ν}).

Hence we have because of Δ_{l_{ν}} ≦ |p_{l_{ν}} − p_{l_{ν}−1}| · tan ∡(u_{l_{ν}−1}, u_{l_{ν}}) ≦ D(C) · \frac{1}{ν} where D(C) is the diameter of C (see (43)), using the definition of the Hausdorff distance,

δ(K, K̂_{ν}) ≦ D(C) · \frac{1}{ν} (ν ∈ ℕ)  (44)

if K resp. K̂_{ν} are the convex bodies in 𝔸^{2} bounded by C ∪ C′ resp. Ĉ_{ν} ∪ C′. In other words (44) may be expressed by

lim_{ν→∞} K̂_{ν} = K  (45)

in the Hausdorff sense. But by (11)

lim sup_{ν→∞} A(K̂_{ν}) = lim sup_{ν→∞} A(Ĉ_{ν}) + A(C′) ≦ A(K) = A(C) + A(C′)

whence finally by (42)

L(𝒯(C)) = lim sup_{ν→∞} A(Ĉ_{ν}) ≦ A(C) = ∫_{C} κ(x)^{\frac{1}{3}} dℋ^{1}(x)

follows. Thus we have shown the wanted inequality (40) which together with the inequality (15) yields the assertion (12) of Theorem 3.1. □

References

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[2] Dolzmann, G.; Hug, D.: Equality of two representations of extended affine surface area. Arch. Math. 65 (1995), 352–356.

[3] Hug, D.: Contributions to affine surface area. Manuscr. Math. 91 (1996), 283–301.

[4] Leichtweiß, K.: Zur Affinoberfläche konvexer Körper. Manuscr. Math. 56 (1986), 429–464.

[5] Leichtweiß, K.: Bemerkungen zur Definition einer erweiterten Affinoberfläche von E. Lutwak. Manuscr. Math. 65 (1989), 181–197.

[6] Riesz, F.; Sz.-Nagy, B.: Vorlesungen über Funktionalanalysis. VEB Deutsch. Verl. d. Wiss., Berlin 1956.

[7] Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge Univ. Press, Cambridge 1993.

[8] Schütt, C.; Werner, E.: The convex floating body. Math. Scand. 66 (1990), 275–290.

Received January 2, 1998