On a formula for sets of constant width in 2d

A formula for smooth orbiforms originating from Euler can be adjusted to describe all sets of constant width in 2d. Moreover, the formula allows short proofs of some laborious approximation results for sets of constant width.


1.
Introduction. Around 1774 Leonhard Euler considered curves, which he called 'curva orbiformis'. A formula describing such curves can be found in §10 of [9]. In a series of papers [10,11,12] in the 1950's Hammer and Sobczyk recalled the formula to construct all 'curves of constant breadth' in the plane. They started from a characterization of 'outwardly simple line families' F = ϕ F ϕ , with F ϕ = y ∈ R 2 ; y · cos ϕ sin ϕ = q (ϕ) for all cos ϕ sin ϕ ∈ S 1 . Since the orientation of a line should play no role, the Lipschitz-continuous function q satisfies q (ϕ + π) = −q (ϕ) for all ϕ.
They use q to define a curve x : [0, 2π] → R 2 of 'constant breadth' by Indeed, for c large enough in (2), that is, with Dq (ϕ) = lim sup h→0 q(ϕ+h)−q(ϕ) h , the function x describes a 'curve of constant breadth' and, moreover, each curve of constant breadth can be obtained this way. The detour through 'outwardly simple line families' and the result spread over three papers does not make it simple for the reader.
Over the years mathematicians have been intrigued by sets of constant width and we would like to mention some of the papers of a more general nature: [18,19,3,7,8,16] and [15]. Also several books dealt with the subject [14,13,6]. Relatively elementary sets of constant width in two dimensions that can be constructed geometricallly, are the so called Reuleaux polygons. These sets are defined by having boundaries consisting of finitely many circular arcs. See the two examples in Fig. 1. The obvious question is, whether one can approximate any set of constant width by a set of constant width that is a Reuleaux polygon. In 1915 Blaschke [4] proved such a result in two dimensions. In 1951 Jaglom and Boltjanski published in Russian the result that all arcs in the approximating Reuleaux polygon can be choosen to have the same curvature. Translations appeared in 1956 in German [14] and in 1961 in English. Their proof is purely geometrical and fills several pages. Also Meissner, who was interested in the applications of sets of constant width, focused on geometry in [18,19] around 1909. The approximations of Tanno in [21] and Wegner in [22] went in the other direction, showing that any set of constant width in two dimensions can be approximated by such sets having a smooth and even analytical boundary. In 2 dimensions there are both the analytical approach originating from Euler as well as the geometrical approach; for example from Meissner. We want to unify these approaches and we will do so by modifying the formula in (2) to one that displays some geometrical quantities. The modified formula is not entirely new. The rudimentary idea seems to be given by Barbier [2] in 1860, and Blaschke [4] sort of mentions it in footnote * * about an alternative way to describe the areaminimisation problem for sets of constant width, but he does not give a derivation or reference.
In Section 3 we will recall the modified formula. Section 4 is devoted to the converse, namely that any set of constant width is described by this formula. Moreover, the formula will be more convenient to give short proofs of the approximations we have just mentioned. This can be found in Section 5.
Let us start by recalling what is today called a set of constant width in R n .
Definition 1.1. A closed, bounded and convex set G ⊂ R n is called 'a set of constant width' L ∈ R + if the following holds for each x ∈ ∂G: 1. for all y ∈ G one has x − y ≤ L; 2. there exists y ∈ ∂G with x − y = L.
An equivalent way of defining a 'set of constant width' G is to say that the width of G in direction ω ∈ S n−1 , given by does not depend on ω. Here S n−1 = {ω ∈ R n with |ω| = 1}. Again this can be reformulated by defining for a closed bounded set G the support function p G : Meissner [18] already knew that a closed, bounded and convex set G is a set of constant width L if and only if Since being convex and of constant width implies being strictly convex, a set of constant width G has for each ω ∈ S n−1 a unique x ω ∈ ∂G such that and ω is an outer normal direction at x ω . Note that x ∈ ∂G maybe reached by several ω ∈ S n−1 : the set ω ∈ S n−1 ; x = x ω is connected, but not necessarily single valued.

2.
The construction by Euler. This corresponding formula, recalling (2), indeed yields the following result: Proof. Before giving the proof we should point out that the width 2r of this set can be expressed in terms of q by (15) and (18) below. Notice that (1) implies that q is 2π-periodic and that ϕ+2π ϕ q (s) ds = 0 for all ϕ.
So x as in (2) is 2π-periodic and moreover, for q ∈ C 1 (R) one finds that For q ∈ C 0,1 (R) we want to use an approximation argument that preserves the conditions in (1) and (3). To do so we consider the function g ∈ L ∞ (R), defined by which is differentiable almost everywhere The function q we recover from q (0) and g through Note that the 2π-periodicity of g shows the 2π-periodicity of q. Moreover, the formula in (9) gives a solution to the integro-differential equation (8). Indeed, Using a mollifier to approximate g by g n we find g n − g L p → 0 for all p and also g n ∞ ≤ g ∞ . Through (9) one finds from g n − g L p → 0 that q n − q ∞ → 0. For c satisfying (3) it follows from g n ∞ ≤ g ∞ that Thus, the smooth function q n inherits the properties of q in (1) and satisfies (3) with the same constant c as for q. Hence we may proceed by assuming that q is smooth. With the property in (11) and the formula in (7) one finds that ϕ → x (ϕ) rotates counterclockwise. Moreover, the outside normal ν (ϕ) at x (ϕ), when |x (ϕ)| = 0, satisfies ν (ϕ) = cos ϕ sin ϕ . The curvature satisfies Being 2π-periodic, rotating to the left and κ > 0 imply that x ([0, 2π]) is the boundary of a well-defined strictly convex domain.
Moreover, (2) implies with (1) and, with p G as in (6), A convex domain for which the expression (12) is constant is a set of constant width.

3.
A modified construction. We will not use (2) directly. Our related formulation was inspired by a video on the educational website [23]. After 3 minutes and 11 seconds this video shows an informative recipe for constructing sets of constant width consisting of circular arcs by rotating sticks of length 2r over a variable center located on the stick with the center at a signed distance a (ϕ) from its center. The video uses a piecewise constant function a. It is not mentioned how the values were chosen. But obviously one cannot take arbitrary values if the curve has to be closed. As mentioned before, the formula was known to Blaschke [3]. Even Barbier seems to explain the construction as early as 1860, but only in words [2, Footnote on Page 279]. The procedure is also described in words in [20]. The video motivated us to ask a) under what condition on a does this construction work, b) if one can generalise the construction to functions a ∈ L ∞ , and c) if all sets of constant width can be constructed in this way. The first two questions are treated in this section and the last one in the next section.
Blaschke must have known the formulation since only then does the remark in [4, Footnote**) on page 505f] showing conditions (14) and (15) make sense. He, however, gives no reference, nor does he seem to make any further use of it.
For q smooth one may translate (2) into (16), with the assumptions in Recipe 3.1 satisfied, throughx So one finds that g from (8) is related to a by a+g = 1 2 π 0 q (s) ds. However, instead of transferring the formula in (16) back to (2) we prefer to give a direct proof that (16) describes the boundary of a set of constant width. Examples relating a andx for r = a ∞ can be found in Fig. 3. Figure 3. Graphs of the functions a in Recipe 3.1 with the corresponding sets of constant width for r = a ∞ .
Proof. Setting Since a satisfies (13), one finds that u is periodic with period 2π. With some straightforward calculations it follows thatx from (16) can be rewritten tõ showing thatx is periodic. The derivative ofx is defined almost everywhere and then satisfiesx Since r ≥ a ∞ , it follows from the sign of the integrand in (19) that u (ϕ) ≥ 0 for ϕ ∈ [−π, π] and hence on R and with (22) thatx rotates counterclockwise. Since the curvature satisfies the set x ([0, 2π]) is the boundary of a well-defined strictly convex domain. From (21), (19) and (13), (14) we also find that By (5) this implies that the enclosed domain is a set of constant width. The conditions u (0) = u (0) = 0 implyx (0) =x 0 . Since u (π) = 2r and u (π) = 0 implỹ x (π) =x 0 − 2r 0 , the claim follows. As mentioned above one can see the construction in (16) of Recipe 3.1 as the rotation of a stick of length 2r over a variable center located on the stick with the center of rotation at a distance a (ϕ) from its center. The inequality in (15) takes care that this center of rotation indeed lies inside the stick. In fact this rotation center reaches r as the smallest distance to one of the ends when r = a ∞ . One also finds that the maximal curvature of the set of constant width is (r − a ∞ ) −1 for r > a ∞ and infinity for r = a ∞ . The minimal curvature is (r + a ∞ ) −1 .
The conditions in (13) and (14) make sure that ends meet. If one wraps a set G of constant width L in a layer of width ε, i.e. if is the Minkowski sum of G with a ball of radius ε, then G ε has constant width L + 2ε. So one may modify r and mollify corners by adding any positive constant ε to find that r + ε also gives a set of constant width.
In the case that a is piecewise constant, one finds a set of constant width with a boundary consisting of finitely many circular arcs. This is nicely illustrated in [23]. 4. From the set of constant width to the function a. In this section we will show the converse of the result in Theorem 3.2. 1. Forx 0 the point on ∂G with maximal x 1 -coordinate andx * the point on ∂G with minimal x 1 -coordinate, one has r = 1 2 x 0 −x * . 2. For κ min ∈ R + the minimal curvature of ∂G and κ max ∈ (0, ∞] the maximal curvature, one has a ∞ = 1 2 κ −1 min − κ −1 max and r ≥ 1 2 κ −1 min + κ −1 max . Remark 1. The Reuleaux triangle in Fig. 1 has r = 1 and One finds κ −1 min = 2 and κ −1 max = 0. Proof. Fixx 0 ,x * and r as stated in the theorem. Since G is a set of constant width, for each ω = cos ϕ sin ϕ ∈ S 1 there is a unique point x ω ∈ ∂G such that x · ω < x ω · ω for all x ∈ ∂G \ {x ω } . Instead of G we consider G ε = B ε (x); x ∈ G , which is again a set of constant width 2r + 2ε. Moreover, if x ε ω denotes the point with Since G ε is convex, Alexandrov's Theorem implies that the curvature of ∂G ε is defined almost everywhere. Moreover, since each x ε ω ∈ ∂G ε lies on the boundary of B ε (x ω ) and some B 2r+ε (x −ω ) with the curvature κ (x ε ω ) of ∂G ε at x ε ω lies in the interval (2r + ε) −1 , ε −1 . See Fig 4. Proof. Fixx 0 ,x * and r as stated in the Proposition. Since G is a set of constant width, for each ω = cos ϕ sin ϕ ∈ S 1 there is a unique point x ω ∈ ∂G such that x · ω < x ω · ω for all x ∈ ∂G \ {x ω } .
Instead of G we consider G ε = B ε (x); x ∈ G , which is again a set of constant width 2r +2ε. Moreover, if x ε ω denotes the point with we find x ε ω = x ω + εω. Since G ε is convex, Alexandrov's Theorem implies that the curvature of ∂G ε is defined almost everywhere. Moreover, since each x ε ω ∈ ∂G ε lies on the boundary of the curvature κ (x ε ω ) of ∂G ε at x ε ω lies in the interval (2r + ε)  Letting ω 0 = cos ϕ0 sin ϕ0 ∈ S 1 be some direction and x ε ω0 be the corresponding boundary point, one may parametrize the boundary locally by Letting ω 0 = cos ϕ0 sin ϕ0 ∈ S 1 be some direction and x ε ω0 be the corresponding boundary point, one may parameterise the boundary locally by − y (t) cos ϕ0 sin ϕ0 , where y is a C 1,1 -function with y (0) = y (0) = 0 and y defined almost everywhere. One finds for the outside normal and curvature: Then the angle ϕ is parameterised by and Φ ε ∈ C 0,1 is differentiable almost everywhere with Φ ε (t) = y (t) 1+y (t) 2 a.e., which is strictly positive and bounded for t small by (26). So locally Φ inv ε exists, lies in C 0,1 , is differentiable almost everywhere and locally near ϕ 0 with a strictly positive and bounded derivative where it is defined. By choosing several ϕ 0 we may parameterise ∂G ε by a function ϕ → x ε (ϕ) : [0, 2π] → ∂G ε that lies in C 0,1 and is diffferentiable almost everywhere.

5.
Applications. By the formula in (16) one may find rather straightforward approximation results for sets of constant width in 2 dimensions. The function a that determines the set is in L ∞ (0, 2π) and can be approximated a) in the L 2 -topology by step functions and b) in the weak L 2 topology by functions that oscillate finitely often between just two values ±r. Moreover, it can be approximated in L 2 c) by functions of class C ∞ using convolution with Friedrichs mollifiers ζ ε , or d) by analytic functions using partial sums of Fourier series. Any approximation of a in L 2 or weak L 2 implies an approximation of the underlying set defined by (16) in the Hausdorff-metric. In the following list of corollaries we approximate a along the lines of a) through d), but we must also make sure that our approximations satisfy (13), (14) and (15).
Blaschke proved in 1915 [4] that two-dimensional sets of constant width 2r can be approximated in C 0 by Reuleaux polygons of constant width 2r + 2ε, that is, by domains having a boundary consisting of circular arcs. Using (16) we can replace the geometric proof by an analytic one. Corollary 1. Let γ ∈ 0, 1 2 . Any two-dimensional set of constant width can be approximated in C 0,γ -sense by a Reuleaux-polygon of constant width.
Our third result concerns an approximation by smooth sets of constant width. On page 60 of [8] Chakerian and Groemer write that Tanno in [21] in 1976 was the first to prove that arbitrary sets of constant width in 2d can be approximated by sets of constant width with smooth boundaries. Using (16) we can give a direct proof. We replace a by a ε = ζ ε * a − 1 2 sin, ζ ε * a sin − 1 2 cos, ζ ε * a cos, where f, g = π 0 f (x) g (x) dx, r by r ε = max (r, a ε ∞ ) andx 0 byx ε 0 + rε−r 0 . The construction in (34) implies that (13) and (14) are satisfied. Since a ∈ L ∞ (R) one finds a ∈ L p (0, 2π) for all p < ∞ and ζ ε * a − a L p (0,2π) → 0 for ε ↓ 0.
One year after Tanno's approximation by C ∞ functions Weber [22] proved that one may approximate sets of constant width by those that have analytic boundary. Also this result is directly obtained using (16). Proof. If a ε is a smooth function approximating a as in Corollary 3, then, by taking a finite Fourier series a N,ε (x) := N k=−N 1 2π e ik· , a ε e ikx one finds an analytic approximation of a ε and hence of a. Note that the coefficients for k ∈ {0, 1} are zero, so a N,ε satisfies (13) and (14). Moreover, a N,ε − a ε ∞ can be made arbitrary small by taking N large. We do not get that a N,ε ∞ equals a ε ∞ = a ∞ , but through replacing r by r + a N,ε − a ε ∞ when r = a ∞ , one can still satisfy (15).
Finally, we wish to remark that our construction provides a simple proof of Barbier's theorem from [2].