Angular measures and Birkhoff orthogonality in Minkowski planes

Let x and y be two unit vectors in a normed plane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}R2. We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankhänel in Beitr Algebra Geom 52(2):335–342, 2011) is an angular measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}μ on the unit circle for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (C)=\pi /2$$\end{document}μ(C)=π/2 whenever C is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.


Introduction
Let K be an origin-symmetric convex body in the plane, that is, a compact convex set with non-empty interior in R 2 , and consider the normed plane (R 2 , · K ), where x K = min {λ > 0 : x ∈ λK} for any x ∈ R 2 . Then K is the unit ball of the norm, and its boundary bd K the unit circle.
Let x, y ∈ bd K be two unit vectors in R 2 . We say that x is Birkhoff orthogonal to y, and denote it by x y, if x K ≤ x + ty K for all t ∈ R. Geometrically, this means that the line through the point x in the direction y supports the unit ball K. In general, Birkhoff orthogonality is not a symmetric relation. Normed planes where Birkhoff orthogonality is symmetric are called Radon planes and the boundaries of their unit balls Radon curves (see the survey [5]). A Borel measure μ on bd K is called an angular measure, if μ(bd K) = 2π, μ(X) = μ(−X) for every Borel subset X of bd K, and μ is continuous, that is, μ({x}) = 0 for every x ∈ bd K. There always exists an angular measure on bd K, such as the one-dimensional Hausdorff measure on bd K normalized to 2π, but an arbitrary angular measure does not necessarily have any relation to the geometry of (R 2 , · K ). A natural problem then is to find angular measures with interesting geometric properties. For instance, Brass [2] showed that whenever the unit ball is not a parallelogram, there is an angular measure in which the angles of any equilateral triangle are equal. This type of angular measure is very useful in studying packings of unit balls [2,8]. Angular measures with other properties have been proposed; see the survey [1,Section 4] for an overview. An angular measure μ is called a B-measure [3] if μ(C) = π/2 for every closed arc C of bd K that contains no opposite points of bd K, and whose endpoints x and y satisfy x y.
The main result of this note (Theorem 1) is a characterization of the normed planes (R 2 , · K ) which admit a B-measure. In order to formulate this theorem, we need to introduce two subsets of bd K.
We call a point x in bd K an Auerbach point, if there is a y ∈ bd K such that x y and y x. In this case we say that x and y form an Auerbach pair. It is well known that Auerbach points exist for any norm [9, Section 3.2]. We denote the set of Auerbach points of K by A(K). Note that A(K) is a closed subset of bd K. We denote the union of open non-degenerate line segments contained in bd K by E(K). This is a strengthening of a result of Fankhänel [3,Theorem 1], where the existence of a B-measure is shown under the condition that A(K)\E(K) contains an arc. (Fankhänel does not explicitly exclude line segments, but it is clear that they have to be excluded, as line segments in A(K) necessarily have measure 0 for any B-measure; see Lemma 3.) We prove Theorem 1 in Section 2, where we also present a smooth, strictly convex, centrally symmetric planar body K such that A(K) is the union of two disjoint copies of the Cantor set and a countable set of isolated points (Example 4). Thus, A(K) is of Lebesgue measure zero and yet, by Theorem 1, there is a B-measure on bd K.
We recall that a subset of a topological space is called perfect if it is closed and has no isolated point. Recall that the support supp(μ) of a Borel measure μ on a topological space X is the set of all x ∈ X such that all open sets containing x have positive μ-measure. It is easy to see that the support of any continuous measure is a perfect set. In the proof of Theorem 1, we rely on the following converse for X = [0, 1]. This is a well-known result holding more generally for any separable complete metric space [6, Chapter II, Theorem 8.1], but for the convenience of the reader we present an explicit construction for this special case in Section 3. It is well known that every non-empty perfect set is uncountable [7, Theorem 2.43] and every uncountable Borel set contains a perfect set [4, Section 6B]. (There is an even larger class, the analytic sets, with this property [4], but we will only need it for F σ sets).

The Auerbach set and B-measure
Given two non-opposite points a, b ∈ bd K, we denote by (a, b) the closed arc from a to b that does not contain any opposite pairs of points. We denote the closed line segment with endpoints a, b ∈ R 2 by [a, b].

Lemma 3. Let K be an origin-symmetric convex body in
Let y 1 , y 2 ∈ bd K such that x y 1 and y 2 x. Then y 1 = y 2 . By possibly replacing y 2 by −y 2 , we assume without loss of generality that y 1 and y 2 are in the same open half plane bounded by the line ox. By possibly replacing x by −x, we may also assume without loss of generality that y 2 and x are in the same open half plane bounded by oy 1 . Let x 1 and x 2 be points on the same side of oy 1 as x such that y 1 x 1 and x 2 y 2 . Then x 1 , x 2 = x. Because y 2 is between x and y 1 , we have that x 1 and x 2 are in opposite open half planes bounded by ox. As above, since μ is a B-measure, μ( (x 1 , x 2 )) = μ( (x, x 1 )) = μ( (x, x 2 )) = 0, hence x / ∈ supp(μ).
Proof of Theorem 1. Let μ be a B-measure on bd K. Then supp(μ) is a perfect set, hence uncountable, and Lemma 3 gives that A(K)\E(K) is uncountable. Conversely, assume that A := A(K)\E(K) is uncountable. We next find an appropriate perfect subset of A and use Proposition 2 to define a B-measure on bd K. We first need to define an auxiliary map φ : A → A(K) by setting φ(x) to be the first y ∈ A(K) in the positive direction along bd K from x so that x y and y x. Then φ is monotone, but not necessarily injective. However, if φ(x 1 ) = φ(x 2 ), then x 1 y and x 2 y, as well as x 1 and x 2 being on the same side of line oy. Thus [x 1 , x 2 ] is a line segment on bd K. Since the set E (K) := {y ∈ bd K : K has more than one supporting line at y} is countable, it follows that for any given y ∈ A(K), there are at most two values of x ∈ A such that φ(x) = y, and there are at most countably many y ∈ A(K) for which there is more than one x ∈ A such that φ(x) = y. In particular, φ is a Borel measurable map.
We next find an appropriate arc (a, b) such that (a, b)∩ A is uncountable. For any x ∈ bd K, let x + denote the first element of A in the positive direction from x, and let x − be the first element of A in the negative direction from x.
Let E(K) denote the union of the closed line segments on bd K. Then E(K) is the union of E(K) with a countable set. Observe that for any p ∈ bd K, the set φ −1 (p) contains at most two points. Thus, φ −1 (E (K)) is countable. Moreover, φ −1 (Ē(K)) is countable, since φ takes at most one value on an open line segment on bd(K). Fix an element and let b = φ(a). Since a / ∈ E (K), the only two points of bd(K) that form an Auerbach pair with a are ±b. Since a / ∈ E(K), the only two points of bd(K) that form an Auerbach pair with b are ±a.
We also obtain that (a, b) ∩ A or (b, −a) ∩ A is uncountable. Thus we may assume without loss of generality that (a, b) ∩ A is uncountable, where φ(a) = b and φ(b) = −a, so it contains a perfect set, and by Proposition 2 there is a continuous probability measure ν on the Borel sets of bd K with supp(ν) ⊆ (a, b) ∩ A. We use ν to define the B-measure as follows. For any Borel set S ⊆ bd K, let Then μ is clearly an angular measure. Showing that μ is a B-measure is somewhat technical, mainly because is not in general a symmetric relation. Let x, y ∈ bd K with x y. We have to show that μ( (x, y)) = π/2. After possibly replacing x by −x and y by −y, we may assume that x ∈ (a, b) ∪ (b, −a) and y ∈ (a, b) ∪ (b, −a). Case 1: x ∈ (a, b). Then either y ∈ (a, b) or y ∈ (b, −a)\{b}. Case 1.1: y ∈ (a, b). There are two cases depending on the relative position of x and y. Case 1.1.1: x ∈ (a, y). Since a / ∈ E(K), we obtain x = a, and since a / ∈ E (K), we obtain y = b. Hence, μ( (x, y)) = π/2 as required. Case 1.1.2: x ∈ (y, b). Since b / ∈ E (K), we obtain y = a, and since b / ∈ E(K), we obtain x = b, and again μ( (x, y)) = π/2. Case 1.2: y ∈ (b, −a)\{b}. In order to show that μ( (x, y)) = π/2, it will be sufficient to show that φ −1 ( (b, y)) equals (a, x) ∩ A up to ν-measure 0. In fact, we show that Vol. 94 (2020) Angular measures and Birkhoff orthogonality 973 First, let p ∈ φ −1 ( (b, y + ))\φ −1 (E(K)). Then φ(p) ∈ (b, y + ) and p ∈ A. Without loss of generality, φ(p) = b, y + , and we want to show that p ∈ (a, x). Clearly, p ∈ (a, b). Suppose that p ∈ (x, b) and p = x. It follows from p φ(p) and x y that φ(p) / ∈ (b, y)\{y}, since otherwise p = x. Therefore, φ(p) ∈ (y, y + ). However, since φ(p), y + ∈ A, we obtain the contradiction φ(p) = y + . Therefore, p / ∈ (x, b)\{x}, and it follows that p ∈ (a, x), which finishes the proof of the ⊆-inclusion of (2).
This completes the proof of Theorem 1.

Example 4.
We present a smooth, strictly convex, origin-symmetric planar body K such that A(K) is the union of two disjoint copies of the Cantor set and a countable set of isolated points. First, let D denote the Euclidean unit disk centered at the origin, and let C be the shorter arc connecting the two points whose angles with the positive x axis are −π/4 and π/4. Let C 0 denote the Cantor set in C. Now, C 0 can be written as where the I n are disjoint open arcs in C.
For each n ∈ Z + , we construct a smooth and strictly convex curve C n connecting the two endpoints of I n with the following properties.
1. C n has the same tangents at the endpoints as D; 2. C n is contained in conv I n ; 3. For any point x of C n , the tangent of C n at x is orthogonal (in the Euclidean sense) to x if, and only if, x is the midpoint or an endpoint of C n .

Consider the bump function
It is well known that Ψ is non-negative, smooth, its support is [−1, 1], and the only points in its support where the derivative is zero are −1, 1 and 1/2. Let the endpoints of I n be (cos α n , sin α n ) and (cos β n , sin β n ), where α n < β n . Let C n be the curve for some small ε > 0. Clearly, C n is a smooth curve, and if ε is sufficiently small, then it is also strictly convex. Moreover, C n satisfies Property 1, as Ψ (−1) = Ψ (1) = 0. If ε is sufficiently small, then C n satisfies Property 2 as well. Finally, to verify Property 3, observe that the tangent of C n is orthogonal to (cos ϕ, sin ϕ) ∈ C n if, and only if, the derivative of ϕ → 1 − εΨ 2 β n − α n ϕ − α n + β n 2 vanishes at ϕ. However, this is only the case at the midpoint and two endpoints of C n . The closed curve is the boundary of a smooth, strictly convex, origin-symmetric planar body K, say. In order to identify the Auerbach points of K, first observe that if x, y ∈ L form an Auerbach pair in K, then x and y are orthogonal in the Euclidean sense. (The converse does not hold, of course.) By this observation and Property 3, for each n ∈ Z + , the only Auerbach point in the relative interior of the arc C n is the midpoint of C n . The same holds for −C n . Again by the observation, all points of C 0 ∪ −C 0 are Auerbach points. Finally, again by the observation, the set of Auerbach points of (bd D\(C ∪ −C)) is the rotation of the previously described set of Auerbach points in (C 0 ∪ −C 0 ) ∪