ON HAMMER'S X-RAY PROBLEM

Transcription

1 ON HAMMER'S X-RAY PROBLEM R. J. GARDNER AND P. McMULLEN ABSTRACT Suppose that two distinct convex bodies in E" have the same Steiner symmetrals in a number of hyperplanes whose normals lie in a plane. Then the directions of these normals are linearly equivalent to a subset of directions of diagonals of a regular polygon. In particular Steiner symmetrals in some four fixed hyperplanes will distinguish between distinct convex bodies. 1. Introduction At the A.M.S. Symposium on Convexity in 1961, P.C. Hammer [3] proposed the following problem: How many X-ray pictures of a convex body must be taken to permit its exact reconstruction? Hammer postulates a convex hole in an otherwise homogeneous solid, and supposes that the intensity at each point of an X-ray picture determines the length of the chord of the body along the corresponding line. There are, in fact, two problems here, according as the pictures are taken from finite points, or from infinity. It is the latter (and, we suspect, easier) problem, involving parallel chords, which we consider here. With this understanding, then, we can state our main result. THEOREM 1. Let Sbea set of directions in a plane in euclidean space E", which is not linearly equivalent to a subset of the directions of diagonals of a regular polygon. Then the X-ray pictures in directions in S distinguish between distinct convex bodies in E". In this context, we shall not regard two convex bodies as distinct if one is a translate of the other. Our convention is that an X-ray picture is labelled with the direction in which it was taken, but contains no indication (such as the image of some fixed point) of the position from which it was taken. Let us emphasize here that our answer to Hammer's question is a partial one, in that, while the procedure of the theorem effectively enables one to distinguish between two different convex bodies, it does not give a method of reconstructing a convex body from its X-ray pictures. This problem has been considered by Giering [2], but he allows himself to choose the directions of his pictures in a way possibly depending on the body itself. The problem of reconstruction, but generalized to not necessarily convex bodies of varying density, has also received much attention recently, in the context of computerized tomography. For details, we refer the reader to the recent articles [5] and [6]. There it is pointed out (among other things) that the invertibility of the Radon transform enables bodies to be reconstructed completely, if all their X-ray pictures (with directions in some plane) are given. But our answer is in quite a different spirit. 2. A reformulation Before discussing the proof of the theorem, we reformulate the idea of an X-ray picture in more geometric terms. Let s be any direction; then the Steiner symmetral of a Received 21 February, [J. LONDON MATH. SOC. (2), 21 (1980), ]

2 172 R.J. GARDNER AND P. MCMULLEN convex body K in direction s is obtained as follows. Let H be the hyperplane through the origin o with normal direction s. If L is a line in direction s, such that Ln K ± 0, let C(K, L) be the segment of L, centred about Ln H, of the same length as Lr\ K. The Steiner symmetral S(K, s) of K in direction s is then the union of these segments C(K, L). As is well known (see [1]), S{K, s) is a convex body, of the same volume as K. From the definition, we have at once: LEMMA 1. direction s. The Steiner symmetral S(K,s) has the same X-ray picture as K in We observe, however, that S(K, s) is immediately determined by the X-ray picture of K in direction s; we shall therefore in future identify the X-ray picture with the Steiner symmetral (in fact, this identification was made in discussing [2]). We further note that all that is needed for the definition is for the hyperplane H not to be parallel to s; this would give us an affine variant of the Steiner symmetral, whose use would underline the essentially affine nature of our problem. 3. The higher dimensional case In this section, we show that we can confine our attention to planar convex bodies. But we begin with a remark that demonstrates that the apparant freedom to translate is illusory. LEMMA 2. The centre of gravity ofk lies on the same line in direction s as that of its Steiner symmetral S{K, s). Unless specifically stated otherwise, we henceforth suppose that all bodies under consideration have their centres of gravity at the origin o. We next have: LEMMA 3. If the theorem holds for n = 2, then it holds for all n. For, suppose that K x and K 2 are convex bodies, with S(K X, s) = S(K 2,s), for each se S. Then for each 2-plane L parallel to lin S, we have S(L n X^, s) = S(L n K 2, s), and so, by assumption, L n K x = L n K 2. Thus A^ = K 2, which proves the lemma. 4. The planar case It is convenient to use Lemma 2 to reformulate Theorem 1 as: THEOREM 2. Let K x and K 2 be distinct convex bodies in E 2 with the same centre of gravity. Let Sbea set of directions, such that S(K {, s) = S(K 2, s)for each se S. Then S is linearly equivalent to a subset of the directions of diagonals of some regular polygon. Before proving the theorem, we give a general example. Let K x be a regular n-gon with centre at o, and let K 2 be obtained from K x by rotation through an angle n/n. Then it is clear that K x and K 2 have the same Steiner symmetrals in each direction parallel to the edges of conv (K x u K 2 ). In fact, if n is even, application of a suitable affinity to K x and K 2 produces non-congruent convex bodies with the same Steiner symmetrals in each of n directions.

3 ON HAMMER'S X-RAY PROBLEM 173 We now proceed to the proof. Let us write C t = bdk, (i = 1,2), where K x and K 2 satisfy the conditions of Theorem 2. Since K x # K 2, we have C x = C 2. We say a (convex) polygon P is an S-polygon if, for each s e S, whenever u is a vertex of P and L is the line through u in direction s, then L either supports P in u alone, or contains another vertex v of P. LEMMA 4. There exists an S-polygon P, with vert P <= u t n C 2. For, since ^ and K 2 have the same area, (int K 1 )\K 2 is a non-empty open set. Let R be any component of (int K 1 f\k 2 ; then R is an open set bounded by an arc of Cj and an arc of C 2, with common (distinct) endpoints, y and z, say. Lets e S. If L is any line in direction s, the open line segments in L, ((intx 1 )\K 2 ) n L, ({\ntkj\kj n L, have the same length. It follows at once that R' = Ufftint K 2 )VM) n L L n i? ^ 0} is a component of (int K 2 j\k l, whose area is the same as that of R. Moreover, the common endpoints y' and z' of the arcs of C l and C 2 which bound R' clearly lie on the lines through y and z in direction s. We can now iterate this procedure, using any sequence of directions in S. Since each region obtained from R by such sequence of operations has the same area as R, and lies in the bounded region K t u K 2, it is clear that the total number of such regions is finite, and the process closes up. If the regions thus obtained are R = R 1,...,R k, and if R { is bounded by arcs of C t and C 2 with common endpoints y t and z,, then {y t, z x,..., y k, z k } is the set of vertices of the required S-polygon (note that some of the points y t and z, may possibly coincide). This proves the lemma. If P is a polygon, the midpoint polygon of P is that polygon M(P) whose vertices are the midpoints of the edges of P. Our next result is due to Reichardt [4]. LEMMA 5. Let P be a convex n-gon, the centroid of whose vertices is the origin o. Write P o = P, and for k = 1,2,..., define P k = sec {%ln)m{p k _ x ). Then the sequence (P 2k ) =0 converges to an ajfinely regular n-gon. For completeness of exposition, we give a brief sketch of the proof. Let the vertices of P be a 0, a lt...,a n _ 1. Then the corresponding vertices of M 2 (P) = M(M{PJ) are a- = i = 0, where all suffixes are taken modulo n. If we regard the a { as complex numbers (with sum 0), then we can represent this as a matrix transformation p' = Cp, where p = (a 0, a x,..., a n _j) T, and C is the circulant matrix C = o o o i i 1 4 2

4 174 R. J. GARDNER AND P. MCMULLEN The eigenvalues of C arecos 2 (jn/n)(j= 0,1,...,rc l),withcorresponding eigenvectors 30 = (l, 'V',..., ( "" lu ) r, where a> = zxp(2in/n). Since {y 0, y lt..., y n _i} is linearly independent, we can write P= I «;J0> for some complex numbers a 0, a l5..., a n _ l5 where a 0 = 0 since a, = 0. If p i = 0 corresponds to P = P o, then to M 2k (P) corresponds Pk = Z a/cos (jnln)) 2k y }. If «. = a n _ ; = 0 for 1 ^j<r, but one of a r or a n _ r is non-zero, then clearly (sec{rn/n)) 2k p k - a r y r + a n _ r y n _ r = g, say, as k -> oo. If Q is the polygon with vertices corresponding to g, then it is easily seen that Q is an affinely regular star-polygon of type {n/r}, and (soc(rn/n)) 2k M 2k (P)^Q. In our case, since each polygon M 2k (P) is convex, Q must also be convex, and we thus have r = 1. This proves the lemma. The proof of the theorem is then completed by the following elementary observation. LEMMA 6. If P is an S-polygon, then so is the midpoint polygon M(P). For, the vector along any diagonal of M(P) is just half the sum of the vectors along the corresponding diagonals of P. It will thus follow, in the notation of Lemma 5, that each polygon F 2 fc ^s an S-polygon, and hence so is the limit polygon Q. But Q is affinely regular, and we thus have Theorem Final remarks Two further consequences can immediately be drawn from the theorem. In each case, we assume S to lie in a plane. THEOREM 3. //S is infinite, the Steiner symmetrals in directions se S distinguish between convex bodies in E". THEOREM 4. Let S = {s l5...,s 4 }. // the slopes of the s { with respect to some coordinate system have a transcendental cross-ratio, then the Steiner symmetrals in directions s l5..., s 4 distinguish between convex bodies in E".

5 ON HAMMER'S X-RAY PROBLEM 175 It is clear that no three directions can satisfy the condition of Theorem 1, and the examples given after the statement of Theorem 2 show that no three directions will suffice. Thus Theorem 4 is a best possible result; in the spirit of this paper, it answers Hammer's original question. We end by raising two questions, which might point the way to future developments. QUESTION 1. When is a given set of convex bodies with hyperplanes of symmetry the set of Steiner symmetrals of some convex body from given directions! QUESTION 2. When is a given set of convex bodies with hyperplanes of symmetry the set of Steiner ymmetrals of some convex body from some directions'! References 1. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper (Springer, 1934). 2. O. Giering, "Bestimmung von Eibereichen und Eikorpern durch Steiner-Symmetrisierungen", Sber. Bayer. Akad. Wiss. Munchen, Math.-Nat. Kl. (1962), P. C. Hammer, "Problem 2", Proceedings of Symposia in Pure Mathematics, volume VII: Convexity. (American Mathematical Society, 1963). 4. H. Reichardt, "Bestatigung einer Vermutung von Fejes Toth", Rev. Roumaine Math. Pures Appl., 15 (1970), L. A. Shepp and J. B. Kruskal, "Computerized tomography: the new medical X-ray technology", Amer. Math. Monthly, 85 (1978), K. T. Smith, D. C. Solmon and S. L. Wagner, "Practical and mathematical aspects of the problem of reconstructing objects from radiographs", Bull. Amer. Math. Soc, 83 (1977), C.S.I.R. (N.R.I.M.S.), Pretoria, South Africa. University College, London.