Fourier transforms and the Funk Hecke theorem in convex geometry

Transcription

1 Journal of the London Mathematical Society Advance Access published July 9, 9 J. London Math. Soc. Page 1 of 17 C 9 London Mathematical Society doi:1.111/jlms/jdp35 Fourier transforms and the Funk Hecke theorem in convex geometry Paul Goodey, Vladyslav Yaskin and Maryna Yaskina Abstract We apply Fourier transforms to homogeneous extensions of functions on S n. This results in complex integral operators. The real and imaginary parts of these operators provide a pairing of stereological data that leads to new results concerning the determination of convex bodies as well as new settings for known results. Applying the Funk Hecke theorem to these operators yields stability versions of the results. 1. Introduction Our interest, in this work, is the determination of convex or star-shaped bodies from information about their projections or sections. This is a topic that has been much studied, especially in the context of centrally symmetric bodies. We refer to the books by Gardner [5] and Schneider [7] for results showing the extent to which centrally symmetric convex bodies are determined by the sizes of their projections and symmetric star bodies are determined by the sizes of their central sections. In recent times there has been a great deal of interest in obtaining similar results in the absence of the symmetry condition. We mention, for example, the work of Groemer [1, 13], Goodey and Weil [8, 9], Schneider [8] andböröczky and Schneider [1]. Here, we use Fourier transforms and the Funk Hecke theorem to obtain further results, similar in nature to those of Schneider and of Böröczky and Schneider. We will show, for example, that a convex body is uniquely determined by knowledge of the average width and average height of each of its shadow boundaries. The average width of the shadow boundary is, in fact, just the mean width of the corresponding projection of the body. For purposes of comparison, we note that Schneider [8] showed that a convex body is determined by the mean width and the Steiner point of each of its projections. We also find similar results involving the average height of the point of intersection of the support hyperplanes of the body with lines through the origin. We use the techniques of Koldobsky [17] to establish these results. The basic idea is to homogeneously extend a function on the sphere S n to R n and then restrict the Fourier transform of the extension back to the sphere. This introduces certain integral operators on the sphere that depend on the degree of homogeneity of the extension. In many cases, the integral operators that arise are those that had previously been studied using spherical harmonics and the Funk Hecke theorem. In order to obtain results for arbitrary, as opposed to symmetric, bodies, we apply these techniques to rather general functions on S n.thus the Fourier transforms obtained are complex valued and provide a natural pairing of integral operators, one acting on even functions and the other on odd functions. We also obtain stability versions of our results and, here, return to the setting of spherical harmonics in order to estimate the asymptotic behaviour of the eigenvalues of the various integral operators. Received 1 November 7. Mathematics Subject Classification 5A, 4B1, 33C55. The second and third authors were supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN

2 Page of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA One of the first systematic approaches to applications of the Funk Hecke theorem to geometric problems occurred in the work of Schneider [6]. He explained, in rather general terms, how the Funk Hecke theorem can be used to establish geometric properties of convex bodies. For example, if all hyperplanes through a point x of a convex body K in R n divide the surface area of K in half, then K is centrally symmetric with respect to x. As described above, the key idea is to associate with such a question a certain integral operator to which one can apply the Funk Hecke theorem. These operators can be thought of as acting on L S n ). In this context, the Funk Hecke theorem shows that the eigenfunctions of the operator are the spherical harmonics and it provides an explicit integral) formulation of the eigenvalues. The desired geometric property is then obtained by showing that all or certain families of) eigenvalues are non-zero. An earlier use of similar techniques occurred in the work of Petty []. The book [11] by Groemer gives an excellent account of many applications of this technique to a variety of problems. More recently, Koldobsky see [17]) pioneered the use of Fourier transforms in establishing related properties of convex bodies. In this situation, the invertibility of the Fourier transform is used to establish the appropriate geometric result. A recent example of this technique can be found in the work of Koldobsky, Ryabogin and Zvavitch [18]. They used Fourier transform techniques to solve the Shephard problem [9] that had earlier been solved by Petty [3] and Schneider [5], using the Funk Hecke theorem. In we describe some of the results of this paper and relate them to earlier works. Section 3 will be devoted to establishing formulas for the various Fourier transforms. In 4 we shall apply our results to show that an arbitrary convex body is determined by knowledge of the average widths and average heights of its shadow boundaries. In 5 we discuss other applications.. Fourier transforms of distributions For g L 1 R n ), we denote the Fourier transform of g by ĝ. Thuswehave ĝx) = gy)e i x,y dy for x R n, R n and here x, y denotes the usual inner product in R n. As mentioned above, a function on S n is extended homogeneously to R n \ o and the Fourier transform of this extension is then restricted to the unit sphere. Specifically, for f C S n )andp C, we denote by f p the homogeneous degree n + p extension of f to R n \ o. Thuswehave ) x f p x) = x n+p f for x o. x Thought of as a distribution, f p has a Fourier transform ˆf p. It is defined, for < Rp <n,by ˆf p,φ = f p, ˆφ = f p x) φy)e i y,x dy dx R n R n for every test function φ from the Schwartz space S of rapidly decreasing, infinitely differentiable functions on R n. We note that we are using, to denote the inner product and the action of a distribution on a test function; the interpretation should always be clear from the context. For fixed φ and f, the mapping p ˆf p,φ is analytic in the region < Rp <n.as explained in [17], this allows us to extend the definition of the distribution ˆf p to other values of p, though the action of the distribution may no longer be by integration. We see that ˆf p is closely related to the Fourier transforms of the distributions r p and r p sgn r on the real line. These were studied in detail in [6, 17] a similar discussion can be found in [14]); here we shall provide a brief overview of the aspects that will be used in what follows.

3 FOURIER AND FUNK HECKE Page 3 of 17 If Rλ >, then we denote by r λ + the distribution on R given by r λ +,φ = r λ φr) dr. This definition can be extended, by regularization, to < Rλ < by setting r λ +,φ = 1 r λ φr) φ)) dr + 1 r λ φr) dr + φ) λ +1. Continuing in this fashion, we extend the definition of the distribution r λ + to all λ C \{,,...}. For a fixed test function φ, the mapping λ r λ +,φ is analytic on C\{,,...}, it has a simple pole at λ = k k N) and the residue there is given by φ k) ) k 1)!. It follows that the r+/γλ λ + 1) form an entire family of distributions with r λ + Γλ +1),φ =) k φ k) ) for λ = k k N). The distributions r λ and r λ sgn r are defined by and r λ,φ = r λ +,φr) + r λ +,φ r).1) r λ sgn r, φ = r λ +,φr) r λ +,φ r)..) Turning to their Fourier transforms, for t R, wehave r λ ) t) = Γλ +1)sin λπ t λ.3) if λ is not an integer; see [6, pp. 173 and 359] or [17, p. 38]. We now examine the extension of these Fourier transforms to integer values of λ. We denote the usual delta function by δ, and thus δ, φ = φ) for any test function φ. Wehave r m ) t) =) m πδ m) t),.4) r m+1 ) t) =) m+1 m + 1)!t m,.5) r m ) t) = ) m+1 π m + 1)! t m+1.6) for m =, 1,,...; see [6, p. 174]. Formula.4) uses the fact that the zero of sinλπ/) at λ =m removes the pole of t λ at λ =m. Formulas.5) and.6) derive from the fact that ) the residues in.1) cancel out for λ of the form, 4,... We have thus extended r λ to a family of distributions that is analytic on λ C\{, 3,...}. In order to deal with λ =, 3,..., we make use of the Laurent series expansion of Γλ + 1) at those values of λ. Forn =1,,...,wehave where Γλ +1)= )n n 1)! [ 1 λ + n ) ] n 1 +Γ 1) +...,.7) ) = n 1

4 Page 4 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA in the case where n = 1. The Laurent series expansion of r λ) at λ = m 1form =, 1,... is r λ ) = δ m)) m)!λ +m +1) + r m) +λ +m +1) r m ln r ) +... Combining these two expansions with.3) yields the following expression for r m) : r m ) ) m t) = 1+ 1 m)! ) m +Γ 1) ln t t m ;.8) see [6, p. 361]. For the distribution r λ sgn r, our results will involve an extra minus sign when compared with [6] since their definition of the Fourier transform is different; see [6, p. 153]. We have r λ sgn r ) t) = iγλ +1)cos λπ t λ sgn t.9) if λ is not an integer; see [6, pp. 173 and 359] or [17, p. 84]; and r m+1 sgn r ) t) =i) m πδ m+1) t),.1) r m sgn r ) t) =i) m+1 m)!t m,.11) r m sgn r ) ) m+1 iπ t) = t m sgn t.1) m)! for m =, 1,,...; see [6, p. 174]. Formula.1) uses the fact that the zero of cosλπ/) at λ =m + 1 removes the pole of t λ sgn t at λ =m + 1. Formulas.11) and.1) follow from the cancellation of the residues in.) for λ =, 3,... Thus, the distributions r λ sgn r ) form an analytic family on C\{, 4,...}. Forλ =, 4,..., we make use of.9) and.7). This gives r m sgn r ) t) = i) m m 1)! ) m 1 +Γ 1) ln t t m.13) for m =1,,...; see [6, p. 361]. In order to describe our results, we introduce some auxiliary functions. For ξ S n,weuse the notation and ξ = {x R n : x, ξ =} ξ + = {x R n : x, ξ > }. For f C S n) and ξ S n, we define F ξ by F ξ t) =1 t ) n 3)/ S n ξ ft ξ+ 1 t ζ) dζ; of course, this definition also makes sense under much weaker conditions, for example, f C S n). The functions F ξ are related to the average values of f over the sections of S n orthogonal to ξ. We recall that, for any such function f and any ξ S n,wehave 1 fθ) dθ = F ξ t) dt;.14) S n see [11] or[], for example. It is important to note that our computations above show that the actions r λ),φ and r λ sgn r ),φ are well defined even if φ is not a test function. It is enough for φ to be sufficiently smooth in a neighborhood of zero and have fast decay at infinity. In particular,

5 FOURIER AND FUNK HECKE Page 5 of 17 these actions are defined for φ = F ξ. Moreover, for Rλ >, the function r λ) + r λ sgn r ),Fξ is an analytic function of λ. For f C S n) and ξ S n, we see in 3.8), 3.9), 3.7) and 3.6) that ˆf ξ) = π u, ξ fu) du i 1 + Γ 1) ln u, ξ ) u, ξ fu) du, S n S n ) 1 ˆf ξ) = Γ 1) ln u, ξ ) fu) du + iπ fu) du fu) du, S n S n S n ξ + 1 ˆf 1 ξ) =π fu) du i t F ξ t) F ξ )) dt, S n ξ 1 ˆf ξ) = t F ξ t) F ξ ) tf ξ) ) dt + fu) du S n ξ iπ f n+1 u),ξ du. S n ξ In addition, if p is a non-positive even integer and f C S n) is odd, it follows from 3.9) that ˆf p ξ) = i) p/ π u, ξ p fu) du,.15) p)! S n ξ + whereas, if p is a negative odd integer and f is even, then 3.8) implies that ˆf p ξ) =) p)/ π u, ξ p fu) du..16) p)! S n For even f, many of the transforms above can be found in [17] and the papers referenced there. The real part of ˆf was established in [18] and used to provide a Fourier transform solution to the Shephard problem [9]. The most significant feature of the imaginary part of ˆf is the hemispherical transform, which plays an important role in some of Schneider s geometrical applications [6] and in Groemer s work [1, 13]. Its real part was calculated by Yaskin and Yaskina in their work on centroid bodies [31]. The real part of ˆf 1 is the spherical Radon transform, which has numerous geometric applications; see [5, 11]. The imaginary part of ˆf 1 is the inverse cosine transform to which we alluded above and will be discussed in more detail in Section 5. The imaginary part of ˆf will be used in Section 4 when we discuss average heights of shadow boundaries of convex bodies. The integral transforms in.15) and.16) appeared in the work of Falconer [4], where he discussed the floating body problem of Ulam. They also play a central role in the study of Lévy representations [19]. More recently, they appeared in [7] in the context of processes of flats in R n andin[4] in the context of Firey projections of convex bodies. 3. Main results First, we recall a well-known relationship between Fourier and Radon transforms. Let φ S be a test function, ξ S n and t R. Then we have that Rφξ; t) = φx) dx x,ξ =t is the Radon transform of φ in the direction ξ at the point t. For every fixed ξ S n,wehave ˆφsξ) =Rφξ; t)) s) for all s R, 3.1)

6 Page 6 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA where, on the right-hand side, we have the Fourier transform of the function t Rφξ; t); see, for example, [17, Lemma.11]. Let f be a continuous function on R n or S n. Denote by f e and f o the even and odd parts of f, respectively, that is, we have f e ξ) = 1 fξ)+f ξ)), fo ξ) = 1 fξ) f ξ)), ξ Rn or S n. We note that the even part of the Fourier transform of f is the Fourier transform of f e,and so there is no ambiguity in the symbols f ˆe and f ˆo. Theorem 3.1. Let f be an infinitely smooth function on the sphere S n and assume that < Rp <n. Then the Fourier transform of the homogeneous extension f p is a homogeneous degree p function whose values on S n are given by ˆf p ξ) = 1 r p) + r p sgn r ),Fξ 3.) for all ξ S n. Remark 3.. In essence, this result can be found in Koldobsky s book [17], although not explicitly. We find it convenient to provide a proof here based on the ideas found there. Remark 3.3. Note that ˆf p can also be written in the form ˆf p ξ) = r p + ),Fξ, but formula 3.) is more convenient since it explicitly provides the even and odd parts of the transform. Proof of Theorem 3.1. We first consider < Rp <1. Let f be an infinitely smooth function on the sphere and let φ be a test function. As noted above, if Rp >, then the action of f p is by integration. The connection, 3.1), between the Radon and Fourier transforms then implies that ˆf p,φ = 1 [ ] fθ) r p ˆφe rθ) dr + r p sgn r ˆφ o rθ) dr dθ. S n Adding the further restriction Rp <1 ensures that the actions of r p) and r p sgn r ) are also by integration, and so we have ˆf p,φ = 1 [ fθ) r p ) ) t) φ e x) dx dt S n x,θ =t ) ] + r p sgn r ) t) φ o x) dx x,θ =t dt dθ.

7 FOURIER AND FUNK HECKE Page 7 of 17 Thus, using.3) and.9), for < Rp <1, we have ˆf p,φ =Γp)cos pπ fθ) t p φ e x) dx dt dθ S n x,θ =t iγp)sin pπ fθ) t p sgn t φ o x) dx dt dθ S n x,θ =t =Γp)cos pπ fθ) x, θ p φx) dx dθ S n R n iγp)sin pπ fθ) x, θ p sgn x, θ φx) dx dθ S n R n =Γp)cos pπ φx) x, θ p fθ) dθ dx R n S n iγp)sin pπ φx) x, θ p sgn x, θ fθ) dθ dx. R n S n Consequently, for < Rp <1, we have that ˆf p is a function, and, for x R n,wehave ˆf p x) =Γp)cos pπ x, θ p fθ) dθ S n iγp)sin pπ x, θ p sgn x, θ fθ) dθ. 3.3) S n We now use.14) tosee,forξ S n, that ˆf p ξ) =Γp)cos pπ 1 t p F ξ t) dt iγp)sin pπ 1 t p sgn tf ξ t) dt. A comparison with.3) and.9) shows that this is the required result in the case where < Rp < 1. To prove the general result, we use the following analytic continuation argument. For < Rp <nand a fixed test function φ, wehave ˆf p,φ = f p x) ˆφx) dx = t p fξ) R n S ˆφtξ) dξ dt. 3.4) n On the other hand, if we extend the right-hand side of equation 3.) froms n to R n \ o with homogeneity p, then its action on φ is given by integration as follows: 1 t n p S n r p ) + r p sgn r),f ξ φtξ) dξ dt. 3.5) Differentiation with respect to p and the remarks in show that 3.4) and 3.5) are analytic functions of p in the region < Rp <n. Since they coincide on the set < Rp <1, we may use analytic continuation to conclude, for each φ and < Rp <n, that The result then follows. ˆf p,φ = x p 1 r p ) + r p sgn r),f x/ x,φx). Our applications will involve real integer values of p, and hence we shall now examine these in more detail. We extend 3.) to these values of p using the regularizations and extensions.4).6),.8) and.1).13).

8 Page 8 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA For p =1, 3,...,.4) and.11) yield ˆf p ξ) =) p)/ πf p) ξ ) i) p)/ p 1)! 1 p t p t sgn t F j ξ t) j! F j) ξ ) dt + j= j= j p j odd whereas, for p =, 4,...,.5) and.1) give 1 p ˆf p ξ) =) p/ p 1)! t p t F j ξ t) j! F j) ξ ) dt + j p j even j!1 + j p) F j) If p =, 3,..., then.6) and.13) show that ˆf p ξ) = )p)/ π p)! + i )p)/ p)! 1 1 t p F ξ t) dt and, for p =,,...,.8) and.1) give ˆf p ξ) = )p/ p)! 1 i )p/ π p)! 1 j!1 + j p) F j) ξ ), 3.6) ξ ) + i)p/ πf p) ξ ). 3.7) ) p +Γ 1) ln t t p F ξ t) dt, 3.8) ) p +Γ 1) ln t t p F ξ t) dt t p sgn tf ξ t) dt. 3.9) Thus far, we have dealt with the Fourier aspects of our results; we now turn our attention to spherical harmonics and the aspects associated with the Funk Hecke theorem. For convenience, we define I p : C S n ) C S n )byi p f = ˆf p. Our geometric applications will involve the determination of f from the knowledge of I p f for some p. Since the Fourier transform including its extensions described above) intertwines the action of the rotation group, it follows from Schur s lemma that I p acts as a multiple of the identity on the spherical harmonics of fixed degree; that is, there is a number λ m n, p) such that we have I p H m = λ m n, p)h m for each spherical harmonic H m of degree m in dimension n. and Lemma 3.4. If < Rp <n, then the eigenvalues λ m n, p) are given by λ m n, p) = p π n/ ) m/ Γm + p)/) Γm + n p)/) λ m n, p) =i p π n/ ) m)/ Γm + p)/) Γm + n p)/) if miseven if m is odd.

9 FOURIER AND FUNK HECKE Page 9 of 17 Proof. Let p be a complex number with < Rp <1. If m is an even integer, then, by 3.3) and the Funk Hecke formula, we have λ m n, p)h m ξ) =I p H m ξ) =Γp)cos π θ, ξ p H m θ) dθ. S n The above integral is easy to calculate; see Koldobsky [16, Lemma 1] or Ournycheva and Rubin [1, 1.4)], for example. This gives λ m n, p) =π n)/ ) m/ Γm + p)/) Γm + n p)/) = p π n/ ) m/ Γm + p)/) Γm + n p)/), Γp)Γ p +1)/) Γp/) cos pπ which is the required result for even m and < Rp <1. We extend it to < Rp <nby analytic continuation. An equivalent formulation of the above result is that, for even m, wehave 1 n 1)κ n t p Pmt)1 n t ) n 3)/ dt = p π n/ ) m/ Γm + p)/) Γp)Γm + n p)/) cospπ/) ; here Pm n denotes the Legendre polynomial of degree m in dimension n, and κ n = π n/ Γ1 + n/) is the volume of the unit ball in R n. Using the standard recursion formulas for the Legendre polynomials see [11, 3.3.6)]) for odd values of m, we see that 1 n 1)κ n t p Pmt)1 n t ) n 3)/ p π n/ ) m+1)/ Γm + p)/) dt = Γp)Γm + n p)/) cosp +1)π/). Now, for odd m, 3.3) implies that Thus we have as required. λ m n, p)h m ξ) =I p H m ξ) = iγp)sin π λ m n, p) = in 1)κ n Γp)sin π 1 = i p π n/ ) m)/ Γm + p)/) Γm + n p)/) S n θ, ξ p sgn θ, ξ H m θ) dθ. t p P n mt) 1 t ) n 3)/ dt if m is odd, In this context, the injectivity of I p is equivalent to the fact that λ m n, p) for all m. Corollary 3.5. p<n. The mapping I p : C S n ) C S n ) is injective for real p with Proof. We note that there is injectivity for a much larger range of p, but this will suffice for our applications. First, we note that, for p<nand a non-negative integer m, neither m + p nor m + n p is a non-positive even integer except for the cases p =, m =1andp =,m=. Thus Lemma 3.4 after a possible analytic continuation) provides the required injectivity result except in these two cases. As explained in, the definitions of I and I are not obtained by analytic continuation, and so it is clear that Lemma 3.4 cannot be applied to these cases.

10 Page 1 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA In order to show that I is injective, we need only show that I H 1 for spherical harmonics H 1 of order 1. The fact that the other harmonics are not annihilated follows from Lemma 3.4. As we saw in the above proof, it will suffice to show that, for any fixed ξ S n, we have I P1 n ξ, ))ξ). To see this, we note that I P1 n ξ, ))ξ) = i 1 + Γ 1) ln u, ξ ) u, ξ du S n = in 1)κ n Γ 1) ln t)t 1 t ) n 3)/ dt. It therefore suffices to show that the above integral is non-zero for all n 3. To this end, we use [1, 4.531) and )] to deduce that Γ 1) ln t)t 1 t ) n 3)/ dt = 1 3 β, n 1 ) 1+Γ 1) + n 1) Of course, the infinite series above is just the finite sum n k=1 ) 1 >. k + n)k +1) if n is odd. Similarly, the injectivity of I is a consequence of the following calculation: 1 Γ 1) ln t)1 t ) n 3)/ dt = 1 1 β, n 1 ) ) Γ 1 1) + n 1). k + n)k +1) The final term above is clearly increasing with n. An easy calculation shows that it is positive for n =3. k= In order to obtain stability versions of various injectivity results, we shall now find estimates on the supremum norm f in terms of the L norm I p f, in the case that f is a difference of support functions of convex bodies. We recall that the support function of a convex body K in R n is defined by h K x) = max ξ K x, ξ, x Rn, and that a convex body is uniquely determined by its support function. For this and other fundamental results in convexity, we refer to the book [7] by Schneider. Here, we denote the Hausdorff distance between the convex bodies K and L in R n by δ K, L). Theorem 3.6. Let K and L be convex bodies in R n, contained in a ball of radius R, and with infinitely smooth support functions. If, for some ɛ, wehave then δ K, L) I p h K ) I p h L ) ɛ, { Cn, p)r n)/n+1) ɛ 4/n p+)n+1) ɛ + R ) n p)/n p+)n+1)) if n>p, Cn, p)r n)/n+1) ɛ /n+1) if n p. Here Cn, p) is a constant that depends only on n and p.

11 FOURIER AND FUNK HECKE Page 11 of 17 Proof. Let f = h K h L and denote its associated spherical harmonic expansion by Q m. m= By Vitale s theorem [3], it is enough to estimate the L -norm of f instead of the supremum norm. We let o denote the gradient operator on S n.itwasshownin[11,..5)] that, if K is contained in a ball of radius R, then, for all ξ S n,wehave o h K ξ) R. 3.1) Furthermore, we have o f = mm + n ) Q m ; 3.11) m=1 see [11, p. 75]. Assume, first, that n>p. Using Parseval s equation, the fact that λ m n, p) andhölder s inequality, we get δ K, L) = f = Q m m= ) = λ m n, p) 4/n p+) Q m 4/n p+) m= ) λ m n, p) 4/n p+) Q m n 4p)/n p+) ) /n p+) m= λ m n, p) Q m n p)/n p+) λ m n, p) 4/n p) Q m ). m= By virtue of Lemma 3.4, we have λ m n, p) Q m = I p f. m= On the other hand, by Lemma 3.4 and Stirling s formula, for fixed n and p, one can see that λ m n, p) 4/n p) Cn, p)m as m tends to infinity. Therefore, using 3.11) and 3.1), we get the following estimate note that the constant Cn, p) may change from line to line): f Cn, p) I p f ) /n p+) Q + Cn, p)ɛ 4/n p+) ɛ + o h K o h L Cn, p)ɛ 4/n p+) ɛ + R ) n p)/n p+). mm + n ) Q m m=1 ) n p)/n p+) ) n p)/n p+)

12 Page 1 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA If n p, then λ m n, p) does not approach zero as m tends to infinity. Thus, there is a constant Cn, p) such that Cn, p) λ m n, p) 1 for all m. Hence we have f = Q m Cn, p) λ m n, p) Q m m= m= = Cn, p) I p f Cn, p)ɛ. We now apply Vitale s theorem to these estimates in order to obtain the desired result. 4. Shadow boundaries The shadow boundary of the convex body K R n in the direction ξ S n comprises those points of the boundary of K that project into a boundary point of K ξ. It follows that it is also the set of all boundary points of K at which there are support lines of K parallel to ξ. If denotes the gradient operator on R n, then h K ξ) exists for almost all ξ S n and is the point of contact with K of the support plane with outward normal ξ; see [7], for example. It follows that, for a strictly convex body K R n, the shadow boundary in the direction ξ S n is { h K u) : u, ξ =}. For such a body, we define the average width of the shadow boundary in the direction ξ to be W K ξ) = h K u),u du. n 1)κ n S n ξ It follows from Euler s formula for homogeneous functions that h K u),u = h K u), and hence we see that W K ξ) is just the mean width of the projection K ξ of K onto ξ. This observation also allows us to extend the notion of the average width of a shadow boundary to arbitrary convex bodies. The average height of the shadow boundary in the direction ξ of a strictly convex body is defined by 1 H K ξ) = h K u),ξ du. 4.1) n 1)κ n S n ξ A fairly immediate geometric application of Theorem 3.1 is the following result. Theorem 4.1. Let K beaconvexbodyinr n with C support function. We have that K is uniquely determined by the average height and average width of all its shadow boundaries. Proof. Let h = h K be the support function of K. It follows from the observations above that the average width of the shadow boundaries of K is a multiple of the spherical Radon transform of h. It is well known that the spherical Radon transform is injective on even functions, and therefore, if we know the mean width of all hyperplane projections of a body K, then we know the even part of the support function of K. In fact, this injectivity result can also be deduced from 3.6) with p = 1, since that result shows that the real part of ˆf 1 ξ) isπf ξ ), which is also a multiple of the spherical Radon transform of f at ξ. Applying this to f = h S n and noting that f n+1 = h K, yields the desired result. In order to recover the odd part of h K, we use Theorem 3.1 with p =. First, we note that K is strictly convex since h K is of class C and therefore the average height of each shadow boundary is well defined. Setting f = h, we see that the odd part of f is determined by the imaginary part of ˆf. It follows from 3.7), with p =, that the imaginary part of ˆf ξ) is

13 FOURIER AND FUNK HECKE Page 13 of 17 given by πf ξ) = π d ft ξ+ 1 t dt ζ) dζ S n ξ = π f n+1 u),ξ du. S n ξ Thus, the odd part of h K is determined by the average heights of the shadow boundaries of K. t= Our next objective is to remove the C hypothesis from Theorem 4.1. This will be achieved by establishing a stability version of the result and will make use of a deep result of Ewald, Larman and Rogers [3]. The shadow boundary of K R n in the direction ξ S n is said to be sharp if the boundary of K contains no line segments parallel to ξ. For such a direction ξ, there is precisely one point of the boundary of K that projects into any given point of the boundary of K ξ. It is then clear that the average height of such a shadow boundary is well defined and that it agrees with our earlier definition 4.1) in the case where h K is a C function. It was shown in [3] that the set of directions ξ for which the shadow boundary is not sharp has a σ-finite n )-dimensional Hausdorff measure in S n. Thus, for almost all ξ S n, the shadow boundary of K in the direction ξ is sharp and the average height of the shadow boundary in the direction ξ exists. Theorem 4.. Let K and L be convex bodies in R n n 5) that are contained in a ball of radius R. If, for some ɛ>, wehave W K W L + H K H L <ɛ, then there is a constant c n, dependent only on the dimension n, such that we have δ K, L) c n R n )/nn+1)) ɛ 4/nn+1)). Proof. The stability aspects of W K correspond to p = 1 in Theorem 3.6, whereas those for H K correspond to p =. If the bodies have infinitely smooth support functions, then the result is a consequence of Theorem 3.6 with p = 1 since this produces weaker estimates than p = ). If K and L are arbitrary convex bodies, then the result follows by approximation. We shall show the idea only for the average heights of shadow boundaries. Recall that H K ξ) orh L ξ) is the average height of the shadow boundary of K or L, respectively, in the direction ξ, ifitexists.letk i or L i be a sequence of infinitely smooth strictly convex bodies approaching K or L, respectively, in the Hausdorff metric; see [7, pp ]. Then, for almost all ξ S n, we also have H Ki ξ) H K ξ)andh Li ξ) H L ξ)as i. However, since the functions H Ki and H Li are uniformly bounded, we have convergence in the L -norm. We also note that we could get stability estimates in lower dimensions using the case p n in Theorem 3.6. For purposes of comparison, we note that Schneider [8] showed that a convex body is determined by the mean width and Steiner points of its projections. He also provided a stability version of this result. 5. Other applications As has been noted by others, the formulations for ˆf p provide alternate proofs of the injectivity properties of certain well-known integral operators. These include the cosine transform R ˆf ),

14 Page 14 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA the spherical Radon transform R ˆf 1 ) and the hemispherical transform, which is the major part of I ˆf. In 4 we made use of the real part of ˆf 1 and the imaginary part of ˆf. In this section we discuss the real and imaginary parts of ˆf 1 and the imaginary part of ˆf. For an odd function f, wehavef ξ ) = for all ξ S n, and hence we have For t, ±1, we have ˆf 1 ξ) = i t F ξ t) =1 t ) / 1 t F ξ t) dt. S n ξ +tξ) 1 fu) du, u, ξ and so ˆf 1 is a sort of inverse cosine transformation. We note that it is the fact that f is odd and smooth that yields the integrability of the integrand on the right-hand side. This gives us an indication of what is an odd analogue of the Radon transform. In the case of support functions f = h K, it is possible to evaluate ˆf 1 without further smoothness conditions. This is already clear for the real part of ˆf 1. To deal with the imaginary part, for ξ S n and u S n \ ξ, we note that 1 h K u) h K u) u, ξ is the average of the heights of the intersection of the support planes with outward normals ±u with the line [ξ] through the origin parallel to ξ. Consequently, it makes sense to refer to 1 1 h K u) h K u) S K ξ) = du dt 5.1) n 1)κ n S n ξ +tξ) u, ξ as the average height of the intersection of support planes to K with [ξ], assuming this integral exists. We now explain briefly why the integral 5.1) exists for all convex bodies K in R n and ξ S n without any smoothness conditions. For t, 1] and ζ ξ S n, we note that the difference quotients h K tξ + 1 t ζ) h K ζ) t are increasing with t. Their limit as t is the directional derivative h K ζ,ξ); see [7, Theorem 1.7.], for example. This, in turn, is the support function of the support set K {x R n : x, ζ = h K ζ)}, evaluated at ξ. It follows that, for fixed ζ, these difference quotients are integrable with respect to t [, 1] and that their integral, as a function of ζ, is bounded. Consequently, S K ξ) exists for all K R n and all ξ S n. A first application of these average heights S K is in the spirit of Schneider s original results [6]. Theorem 5.1. AconvexbodyK is centrally symmetric if and only if S K or H K )isa linear function. Equivalently, S K ξ) = x, ξ or H K ξ) = x, ξ ) if and only if K is centrally symmetric with respect to x. Proof. For an arbitrary convex body K R n,welet m= Q m be the spherical harmonic expansion of h K. Then K is centrally symmetric if and only if Q m = for all odd m 1.If f = h K on S n, then ˆf 1 has the spherical harmonic expansion m= λ mn, 1)Q m on S n. It follows, from Lemma 3.4, that λ m n, 1) for all m, andsok is centrally symmetric if and only if λ m n, 1)Q m = for all odd m 1. Using 5.1), we see that S K has the spherical

15 FOURIER AND FUNK HECKE Page 15 of 17 harmonic expansion m odd Rλ mn, 1)Q m. Hence K is centrally symmetric if and only if S K = Rλ 1 n, 1) x, = nκ n x, for some x R n. In this case, K is centrally symmetric with respect to the point x R n.the proof for H K is analogous. Using both the real and imaginary parts of ˆf 1, we have the following theorem. Theorem 5.. AconvexbodyK R n is completely determined by knowledge of the integrals 1 h K u) h K u) h K u) du and du dt S n ξ S n ξ +tξ) u, ξ for all ξ S n.thus,k is determined by knowledge of the mean widths of all projections onto hyperplanes and of all the average heights S K ξ). Theorem 3.6 again provides stability versions of these results. Corollary 5.3. Let K and L be convex bodies contained in a ball of radius R in R n n 3). If, for some ɛ>, wehave S K S L <ɛand the L distance between the mean widths of the projections of K and L is bounded by ɛ, then there is a constant c n, dependent only on the dimension n, such that δ K, L) c n R n )/nn+1)) ɛ 4/nn+1)). Corollary 5.4. For n 3, assume that K is a convex body in R n and x R n,andchoose R so that K is contained in the ball with centre o and radius R, and so that x R. Assume that, for some ɛ>, wehave S K nκ n x, ɛ. Then there exists a centrally symmetric body C R n centred at x and such that δ K, C) c n R n )/nn+1)) ɛ 4/nn+1)). Proof. Let m= Q m be the spherical harmonic expansion of h K. We note that m even Q m is the spherical harmonic expansion of the support function of the central symmetrand of K. Let C be that symmetrand translated by x. Now,S C = nκ n x, and the projections of K and C have the same mean widths. Hence the result follows from Corollary 5.3. The above results are somewhat related to recent work of Böröczky and Schneider [1] and of Schneider [8]. Their work involved the determination of a function f CS n ) from certain integrals. The integrals they used were the spherical Radon transforms of f and of the products ξ, f for each ξ S n. The former integrals determine the even part of f, and the latter determine the even part of ξ, f or, equivalently, the odd part of f. The combination of the two provides a unique determination of the function f. Their results had elegant geometric interpretations and allowed them to show, on the one hand, that convex bodies are determined by the mean width and Steiner points of their projections and, on the other hand, that a star body is determined by the volumes and centroids of its central sections. In fact, the major part of their work was devoted to establishing stability versions of these results for convex bodies. In this context, we note that knowledge of the spherical Radon transform Rf of f CS n ) is equivalent to knowing that S n gu)rf)u) du for all even g CS n )

16 Page 16 of 17 PAUL GOODEY, VLADYSLAV YASKIN AND MARYNA YASKINA or, equivalently, that S n gu)fu) du for all even g CS n ). Analogously, knowledge of the spherical Radon transforms of all products ξ, f is equivalent to knowing that S n gu) ξ,u fu) du for all even g CS n ) and all ξ S d. However, it follows from 3.8), with p =, that the odd part of f is determined by knowledge of S n Γ 1) ln u, ξ ) u, ξ fu) du for all ξ S n. Unfortunately, we have not found any natural geometric interpretation of these integrals when f is a support function. We note that our estimates in Theorems 3.6 and 4. and therefore those in Corollaries 5.3 and 5.4 could be improved using recent techniques of Kiderlen [15] that, in turn, build on ideas of Campi []. We would like to thank them for their helpful comments. References 1. K. Böröczky and R. Schneider, Stable determination of convex bodies from sections, Studia Sci. Math. Hungar., toappear.. S. Campi, Recovering a centered convex body from the areas of its shadows: a stability estimate, Ann. Mat. Pura Appl. IV ) G. Ewald, D. Larman and C. A. Rogers, The direction of the line segments and the r-dimensional balls on the boundary of a convex body in Euclidean space, Mathematika ) K. J. Falconer, Applications of a result on spherical intregration to the theory of convex sets, Amer. Math. Monthly ) R. J. Gardner, Geometric tomography, nd edn Cambridge University Press, Cambridge, 6). 6. I. M. Gelfand and G. E. Shilov, Generalized functions, vol. 1: properties and operations Academic Press, New York, 1964). 7. P. Goodey and R. Howard, Processes of flats induced by higher dimensional processes II, Contemp. Math ) P. Goodey and W. Weil, Directed projection functions of convex bodies, Monatsh. Math ) P. Goodey and W. Weil, Average section functions for star-shaped sets, Adv. in Appl. Math. 36 6) I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series and products, 7th edn Academic Press, Amsterdam, 7). 11. H. Groemer, Geometric applications of Fourier series and spherical harmonics Cambridge University Press, New York, 1996). 1. H. Groemer, On the girth of convex bodies, Arch. Math ) H. Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math ) L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, nd edn, Grundlehren der Mathematischen Wissenschaften 56 Springer, Berlin, 199). 15. M. Kiderlen, Stability results for convex bodies in geometric tomography, Indiana Univ. Math. J. 57 8) A. Koldobsky, The Busemann Petty problem via spherical harmonics, Adv. Math ) A. Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs 116 American Mathematical Society, Providence, RI, 5). 18. A. Koldobsky, D. Ryabogin and A. Zvavitch, Projections of convex bodies and the Fourier transform, Israel J. Math ) P. Lévy, Théorie de l addition de variable aléatoires Gauthiers-Villars, Paris, 1937).. C. Müller, Spherical harmonics Springer, Berlin, 1966). 1. E. Ournycheva and B. Rubin, Composite cosine transforms, Mathematika 5 5) C. M. Petty, Centroid surfaces, Pacific J. Math ) C. M. Petty, Projection bodies, Proc. Coll. Convexity, 1965, Copenhagen Kobenhavns Univ. Mat. Inst., Copenhagen, 1967)

17 FOURIER AND FUNK HECKE Page 17 of D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J. 53 4) R. Schneider, Zu einem Problem von Shephard über die projektionen konvexer Körper, Math. Z ) R. Schneider, Über eine Integralgleighung in der Theorie der konvexen Körper, Math. Nachr ) R. Schneider, Convex bodies: the Brunn Minkowski theory Cambridge University Press, Cambridge, 1993). 8. R. Schneider, Stable determination of convex bodies from projections, Monatsh. Math. 15 7) G. C. Shephard, Shadow systems of convex bodies, Israel J. Math. 1964) R. Vitale, L p metrics for compact convex sets, J. Approx. Theory ) V. Yaskin and M. Yaskina, Centroid bodies and comparison of volumes, Indiana Univ. Math. J. 55 6) Paul Goodey Department of Mathematics University of Oklahoma Norman, OK 7319 USA pgoodey@math ou edu Vladyslav Yaskin and Maryna Yaskina Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB Canada T6G G1 vladyaskin@math ualberta ca myaskina@math ualberta ca