Measuring Ellipsoids 1

Transcription

1 Measuring Ellipsoids 1

2 Igor Rivin Temple University 2

3 What is an ellipsoid? E = {x E n Ax = 1}, where A is a non-singular linear transformation of E n. Remark that Ax = Ax, Ax = x, A t Ax. The matrix Q = A t A is a positive definite matrix, with eigenvalues λ 1,..., λ n, whose (positive) square roots σ 1,..., σ n are the so-called singular values of A. Their geometric significance is that the semi-axes of E are the quantities a i = 1/σ i. After an orthogonal change of coordinates, we can write E = {x E n n x 2 i λ i = i=1 n i=1 x 2 i σ2 i = n i=1 x 2 i a 2 i = 1}. 3

4 Volume of an Ellipsoid vol E = κ n det A = κ n n a i, i=1 where κ n is the volume of the unit ball in E n. 4

5 k-th Mean Curvature m k ( K, x) ( ) n 1 m k k (x) = s k (k 1 (x),..., k n 1 (x)). k-th integral mean curvature M k M k = K m k (x)da, 5

6 Steiner s Formula for Area of Parallel Surface (at distance ρ) vol n 1 K ρ = n 1 k=0 ( ) n 1 M k k ( K)ρ k, The volume of a neighborhood of thickness ρ vol Kρ = vol K + ρ 0 vol n 1 K τ dτ. 6

7 Another formula for M k Let G n,r be the Grassmanian of affine r-planes in E n, with a suitably normalized invariant measure µ. Then µ ( {L r G n,r L r K } ) = ω n 2... ω n r 1 M (n r)ω r 1... ω r 1 ( K), (1) 0 where ω k is the surface area of the unit sphere in E k+1. 7

8 Theorem 1 (Archimedes axiom). M( K) is monotonic under inclusion. That is, if K 1 K 2, then M i ( K 1 ) M i ( K 2 ). The inequality is strict if K 2 \K 1 has non-empty interior, and i < n 1, where n is the dimension of the ambient Euclidean space. 8

9 Integral Mean Curvature for Polytopes Let P be a convex polytope in E n. ( ) n 1 M i i ( P) = α ( f )vol n i 1 f, codim. i faces f of P (2) where α ( f ) is the exterior angle at f, described as follows: Consider the Gauss map, which maps each point p of P to the set of outer normals to the support planes to P passing through p. The image of all of P will be the unit sphere S n 1 E n, and the combinatorial structure of P will induce a dual cell decomposition C of S n 1, in particular, the image of a codimension-i face f of P will be an i-dimensional totally geodesic face f of C. The i-dimensional area of that face is the exterior angle at f. 9

10 Polyhedra which tile space Suppose that translates of P tile E n, so that P is a fundamental domain for a free action of a group G of translations on E. In particular, G acts on P, preserving the combinatorial structure. Let the quotient by P G. Then ( ) n 1 M i i = ω i vol n 1 i f. codim. i faces f of P G In particular, if P is a rectangular parallelopiped: P = [0, l 1 ] [0, l 2 ] [0, l n ], then we obtain M i (P) = ω i s n 1 i (l 1,..., l n ). 10

11 Inscribed and Circumscribed Cubes The cube C i with vertices ( ) ±1, ±1,, ±1 n n n inscribed in the unit sphere, while the cube C o with vertices (±1, ±1,, ±1) is circumscribed around the unit sphere. is If E is an ellipsoid with semi-axes a 1,..., a n, the preimages of P i of C i and P o of C o are, respectively, inscribed and circumscribed (rectangular) parallelopipeds. From the Archimedean theorem, it follows that the mean curvatures of E are contained between those of P i and P o. 11

12 Which means that: M i (P o ) ( n ) n 1 i = M i ( P n ) M (E) M i (P o ), or ( ) n 1 i 2 ωi s n 1 i (l 1,..., l n ) ( n n 1 ) M i (E) i and M i (E) 2 n 1 iω is n 1 i (l 1,..., l n ) ). ( n 1 i In three dimensions, the upper bound of is sharp, as shown by examining ellipsoids with l 1 = 1, l 2 = 1, l 3 1. The lower bound is not sharp; the sharp bound is given by Polya and Szegö in a 1948 paper. 12

13 Area of the Equidistant Surface Theorem 2. Let f (ρ) = 1 n n (ρ + 2l ρ i ) ρ n 2 n l i. i=1 Then the area of the equidistant surface E ρ is bounded as follows: i=1 c n f (ρ) vol n 1 E ρ C n f (ρ), where C n /c n ( n ) n 1 Γ ((n + 1)/2) 2π (n+1)/2. (estimate for volume follows immediately). 13

14 Lattice Points Consider an ellipsoid E, and consider the number N(e) of points of the integer lattice in E. How do we estimate (E) = N(E) vol E? By a fairly obvious argument argument, (E) is dominated by the volume of a tubular neighborhood of radius (n) around E. We have: (E) C n n 0 f (ρ)dρ. (3) 14

15 John ellipsoid It is a well-known theorem of Fritz John that for any convex body K, there exists an ellipsoid E K, such that E K /n K E K for centrally-symmetric K, E K /n can be improved to E K / n. We can estimate the symmetric functions of the semi-axes (and hence the semi-axes themselves) of the John ellipsoid in terms of the mean curvature integrals of K, and vice versa. 15

16 Can we do better? 16

17 Cauchy s formula Let K be a convex body in E n. Let u S n 1 be a unit vector, and let us define V u (K) to be the (unsigned) n 1-dimensional volume of the orthogonal projection of K in the direction U. Cauchy s formula then states that V n 1 ( K) = n 1 ω n 2 V u(k) dσ S n 1 = (n 1) ω n 1 ω V u(k) dσ, n 2 S n 1 where dσ denotes the standard area element on the unit sphere. 17

18 Projected areas of an Ellipsoid In the case where K = E is an ellipsoid, given by n E = {x E n i=1 q 2 i x2 i = 1} there are several ways of computing V u. The result is: ( ni=1 u 2 ) V u (E) = κ n 1 i q2 i ni=1 q i. (4) 18

19 Since V n (E) = κ n ni=1 q i, we can rewrite Cauchy s formula (4) for E in the form: R(E) def = V n 1( E) V n (E) = n S n 1 n i=1 u 2 i q2 i dσ, (5) where R(E) is the isoperimetric ratio of E. 19

20 Amazing, but true! Theorem 3. The ratio R(E) is a norm on the vectors q of inverse lengths of semiaxes (q = (q 1,..., q n ).) Proof. The integrand in the formula (5) is a norm. Corollary 4. There exist constants c n,p, C n,p, such that c n,p q p R(q) C n,p q p, where q p is the L p norm of q. 20

21 How can we compute integrals ove S n? Theorem 5. Let f (x 1,..., x n ) be a homogeneous function on E n of degree d (in other words, f (λx 1,..., λx n ) = λ d f (x 1,..., x n ).) Then ( ) ( ) n + d n Γ 2 f dσ = Γ E ( f (X S n 1 2 1,..., X n ) ), where X 1,..., X n are independent random variables with probability density e x2. 21

22 Application to Ellipsoids R(E) = n S n 1 = n Γ ( n 2 ) Γ ( n+1 2 n u 2 i q2 i dσ )E i=1 ( ) q 2 1 X q 2 n X n where X i is a Gaussian with variance 1/2., 22

23 A class of Hypergeometric Functions First, we need a definition: Definition 6. Let a, b 1,..., b n, c, x 1,..., x n be complex numbers, with x i < 1, i = 1,..., n, Ra > 0, R(c a) > 0. We then define the Lauricella Hypergeometric Function F D (a; b 1,..., b n ; c; x 1,..., x n ) as follows: F D (a; b 1,..., b n ; c; x 1,..., x n ) = Γ(c) Γ(a)Γ(c a) 1 0 u a 1 (1 u) c a 1 n (1 ux i ) b i du. (6) i=1 We also have the series expansion: F D (a; b 1,..., b n ; c; x 1,..., x n ) = (a) ni=1 m1 + +m n (b i ) mi m 1 =0 m n =0 valid whenever x i < 1, i. (c) m1 + +m n n i=1 x m i i m i!, (7) 23

24 Now, we can write ( ) R(E) = n Γ2 n 2 Γ ( ) α 2 n+1 2 n q 2 ( j 2 F D 1/2; η 1j,..., η nj ; n + 1 ) 2 ; 1 αq2 1,..., 1 αq2 n j=1 where η ij = 1/2 + δ ij, and α is a positive parameter satisfying 1 αq 2 j < 1., (8) 24

25 Low Dimensions In two dimensions, the circumference of an ellipse with semiaxes a b is: ( ) C(a, b) = 4aE π/2, 1 b 2 /a 2. In three dimensions, the surface area of an ellipsoid with semiaxes a b c is given by ( A(a, b, c) = 2π c 2 bc 2 ) + a a 2 c 2F(φ, m) + b 2 c 2 E(φ, m), where m = a2 (b 2 c 2 ) b 2 (a 2 c 2 ), and φ = arcsin 1 c2. a 2 25

26 Law of Large Numbers Theorem 7. Let q 1,..., q n,... be a sequence of positive numbers such that ni=1 q 4 i lim n ( ni=1 q 2 ) = 0. 2 i Let E n be the ellipsoid in E n with semiaxes a 1 = 1/q 1,..., a n = 1/q n. Then lim n Γ ( ) n+1 2 Γ ( ) n 2 R(E n ) n 12 ni=1 q 2 i = 1. 26

27 Corollary 8. Let a 1,..., a n,... be such that 0 < c 1 a i /a j c 2 <, for any i, j. Let E n be the ellipsoid with major semi-axes a 1,..., a n. Then lim n Γ ( n+1 2 Γ ( ) n 2 ) n R(E n ) 1 2 ni=1 1 a 2 i = 1. 27

28 Finite Dimensions We know that R(E) is a norm on the vector q = (q 1,..., q n ) let us agree to write q R def = R(E) n = S n 1 n q 2 i x2 i dσ. i=1 where q is the vector of inverses of the major semi-axes of E. We know that for any p > 0, c n,p q p q R C n,p q p, for some dimensional constants c n,p, C n,p. we will give good estimates on the constants c n,2 and C n,2. How good? 28

29 3Γ ( n 2 ) 2Γ ( n+1 2 ) q R q ( ) Γ n2 ( ). (9) πγ n

30 First Basic Inequality Lemma 9. Let F(x) be a probability distribution, and let M a (F) def = E F ( x a ) denote the absolute moments of F (we will abuse notation in the sequel by referring to the absolute moments of a random variable as well as those of its distribution function). Further, let 0 β α. Then M β (F) 1 + M α (F). 30

31 Second Basic Inequality n n S n 1 i=1 q 2 i u2 i dσ i=1 q 2 i S n 1 u 1 dσ. (holds by concavity of square root). 31