Applications of the Brascamp-Lieb and Barthe inequalities Exercise 12.1 Show that if m Ker M i {0} then both BL-I) and B-I) hold trivially. Exercise 12.2 Let λ 0, 1) and let f, g, h : R 0 R 0 be measurable functions satisfying that hx λ y 1 λ ) fx) λ gy) 1 λ, for all x, y R 0. Show that 0 hx) dx 0 ) λ 1 λ fx) dx gx) dx). 0 Hint: Consider the function F x) = fe x )e x, and analogously for g and h. Exercise 12.3 Let c i > 0 and v i S n 1, 1 i m, such that m c i v i v i ) = I n, and let α i > 0, 1 i m and 1 p <. For K K0 n with x K = m v i, x p ) 1/p we know that Show that the inequality is best possible. ) n Γ 1 + 1 vol K) 2 n p ) Γ 1 + n p m α i ci α i ) ci /p. Exercise 12.4 Let A, A i R n n, 1 i m be positive definite symmetric matrices and let λ i > 0, 1 i m, with m λ i = 1. i) Show that ) π e x,ax n 1/2 dx =. R n det A Hint: Recall that for a positive definite symmetric matrix A R n n, the ellipsoid given by EA, 0) = { x R n : x, Ax 1 } has volume vol EA, 0) ) = vol B n )/det A) 1/2. ii) Using the above identity and BL-I), show that π n/2 det m λ ia i ) ) m π n/2 λi. det Ai Discussion: 26.01.2009 List 12
John s ellipsoids Exercise 11.1 Let P B n be a polytope with m facets. Show that its inradius satisfies rp ) c log m)/n, for a suitable constant c. Exercise 11.2 Let u 1,..., u m S n 1, and let P = { x R n : u i, x 1, 1 i m }. Let r 1 be such that µ n 1 rs n 1 P ) 1/2. Show that vol P ) r n /2)vol B n ). Exercise 11.3 For u i and P as in the previous exercise, show that for a suitable constant c. vol P ) ) n n/2 c vol B n ), log m Exercise 11.4 Let v S n 1, t [0, 1] and let U = { u S n 1 : v, u t }. Show that µ n 1 U) 1 4e nt2 /4 and, in particular, for t 4/ n it holds µ n 1 U) 0.9. Hint: Assume, without loss of generality, that v = e n and use the Measure concentration theorem for the sphere. Discussion: 19.01.2009 List 11
John s ellipsoids Exercise 10.1 How is the volume of the ball distributed?) Let B n denote the n-dimensional ball of volume 1. i) Using Stirling s formula, show that a section of B n through the center maximal section) has n 1)-dimensional volume about e. ii) Show that any parallel section to the previous one, at distance x from the center, has n 1)- dimensional volume about fx) = e exp πex 2 ). Exercise 10.2 0-symmetric polytopes as sections of the cube) Let v 1,..., v m R n \{0} and let P = { x R n : 1 v i, x 1, 1 i m } be a 0-symmetric polytope. Prove that P can be obtained as the intersection of an m-dimensional cube m > n) with an n-dimensional plane. Exercise 10.3 Let V R n. Prove that conv V v V 1 2 v + 1 ) 2 v B n. Exercise 10.4 Use the previous exercise in order to prove that if v i P = conv {v 1,..., v m }, then vol P ) m 2 n vol B n). B n, i = 1,..., m and Discussion: 12.01.2009 List 10
John s ellipsoids Exercise 9.1 Let K, L K n 0. i) Prove that the Banach-Mazur distance is defined as d BM K, L) := inf { λ > 0 : T GLn, R) with K T L λk }. ii) Show that d BM K, L) = d BM K, L ). Exercise 9.2 Let K, L K n and let ) 1/n vol K) αk, L) = max { vol t + T L) : t + T L K, T GLn, R), t R n}. Determine an upper bound for the ratio αk, L)/d BM K, L). Exercise 9.3 Let S R n be an n-dimensional simplex. Show that RS)/rS) n with equality if and only if S is regular. Exercise 9.4 Show Jung s theorem: Let K K n. Then n RK) DK) 2n + 1), and equality holds if and only if K contains a regular n-simplex of diameter DK). Use the following lemma: Let x i R n, 1 i m, with x i x j D, 1 i, j m. Then there exists x R n with x i x D n/2n + 1)), 1 i m, and equality holds if and only if {x 1,..., x m } contains the vertices of a regular simplex of diameter D. Discussion: 15.12.2008 List 9
John s ellipsoids Exercise 8.1 Let K R n be a closed convex set and let F be a face of K with 0 relint F. The support normal) cone of the face F is defined as the support normal) cone of K at any point of the relative interior of F. Let C = {λu : u K, λ 0} be the support cone of F. Show that C is the normal cone of K at F, i.e., C = cl { v R n : v, x = 0 for all x F, and v, x < 0 for all x K\F }. Exercise 8.2 Let K, L K n, and for x bd K we write SK, x) and NK, x) to denote the support cone and the normal cone of K at x, respectively. Show that: i) If C 1, C 2 R n are two closed convex cones with apex the origin), then C 1 C 2 ) = C 1 +C 2. ii) For x K and y L it holds NK + L, x + y) = NK, x) NL, y) and SK + L, x + y) = SK, x) + SL, y). Exercise 8.3 Let K K n. Show that if B n is the minimum volume ellipsoid containing K, then 1/n)B n K. Exercise 8.4 Let K K n and let B n be the maximum volume ellipsoid contained in K. Let λ i > 0 and u i S n 1 bd K, 1 1 m, the values and points defined by John s theorem. Show that: i) For any x, y K it holds ii) DK) 2nn + 1). m λ i 1 ui, x ) 1 u i, y ) = n + x, y. Discussion: 08.12.2008 List 8
John s ellipsoids Exercise 7.1 Let K, L, M K n. Prove the following properties for the Banach-Mazur distance: i) d BM K, L) = d BM L, K). ii) d BM K, L) 1 and d BM K, x + T K) = 1 for all x R n and T GLn, R). iii) d BM verifies the multiplicative triangular inequality, i.e., d BM K, M) d BM K, L) d BM L, M). Exercise 7.2 If B n is the minimum volume ellipsoid containing K, then 0 int K. Exercise 7.3 For a convex body K K n let { } Ft + AK) n νk) = inf : A GLn, R), t Rn. vol t + AK) n 1 Find upper and lower bounds depending only on the dimension) for νk). Exercise 7.4 Determine a minimal set of vertices V of the 3-dimensional cube C 3 such that the minimum volume ellipsoid containing conv V coincides with the minimum volume ellipsoid of C 3. Discussion: 01.12.2008 List 7
John s ellipsoids Exercise 6.1 Show that C pds = {x A R nn+1)/2 : A is positive definite} is a convex cone with apex the origin and with interior points, i.e., it has dimension nn + 1)/2. Exercise 6.2 Show that Cpds 1 = {x A C pds : det A 1} is a closed and strictly convex set which is smooth, i.e., for each boundary point of Cpds 1 there exists a uniquely determined supporting hyperplane. Exercise 6.3 Let S n be the regular simplex of inradius 1 and let B n S n. i) Determine its volume, surface area, circumradius and diameter. ii) Let u 1,..., u n+1 S n 1 bd S n. Show that n+1 u i = 0 and for any x R n it holds x = n/n + 1) n+1 x, u i u i. iii) Show that B n is the maximum volume ellipsoid contained in S n. Exercise 6.4 Let K K n. If x + α B n is the circumball of K then K x + α B n and the center x conv { x + αs n 1 ) bd K }. Discussion: 24.11.2008 List 6
Brunn-Minkowski type inequalities Exercise 5.1 Let C = [0, 1] n. Show that vol k C L) 1 for any k-dimensional plane L R n. Exercise 5.2 Show that the Steiner symmetral of a simplex with respect to a hyperplane orthogonal to any of its edges is also a simplex. Exercise 5.3 Let A i R n n be positive definite symmetric) matrices and let λ i > 0, 1 i m, such that m λ i = 1. Then m ) 1/n det λ i A i m m λ i det A i ) 1/n det A i ) λi/n. Equality holds in the first inequality if and only if A i = µ i A 1 for all i = 2,..., m. Equality holds in the second inequality if and only if det A 1 = = det A m. Exercise 5.4 For K K n with interior points, let F K, x), x K, be a non-negative real-valued function, continuous in K and x, such that F µk + a, µx + a) = µ m F K, x) for some m > 0 and for all µ > 0 and a R n. We also define GK) = K F K, x) dx. Suppose that log F λk + 1 λ)l, λx + 1 λ)y ) λ log F K, x) + 1 λ) log F L, y) whenever x K and y L, for K, L K n, and for 0 λ 1. Prove the following statements: i) GµK) = µ m+n GK) for all µ > 0. ii) G λk + 1 λ)l ) GK) λ GL) 1 λ. Discussion: 17.11.2008 List 5
Brunn-Minkowski type inequalities Exercise 4.1 Show that 1 0 n1 t) n 1 t n dt = n!)2 2n)! Exercise 4.2 Let K K n 0 and let a Rn. Show that vol n 1 K Ha, b) ) is maximal for b = 0. Exercise 4.3 Let a i, b i 0. Then n 1/n n ) 1/n n ) 1/n a i + b i )) a i + b i, and equality holds if and only if a i = λ b i, 1 i n. Exercise 4.4 Hölder-inequality) If f, g : R n R 0 are integrable functions with non-vanishing integrals) and p, q > 1 such that 1/p + 1/q = 1, then ) 1/p ) 1/q fx) gx) dx fx) p dx gx) q dx. R n R n R n Hint: Apply the AGM-inequality to the numbers a 1 = fy) p R fx) p dx n and a 2 = gy) q R n gx) q dx. Discussion: 10.11.2008 List 4
Symmetrisations Exercise 3.1 Prove that the regular simplex has maximal volume among all simplices of circumradius 1. Exercise 3.2 Prove the following statements: i) Any zonotope is centrally symmetric with respect to some point. ii) Any planar centrally symmetric polygon is a zonotope. iii) Let n 3 and let P P n be a full-dimensional polytope. Then P is a zonotope if and only if all its 2-faces are centrally symmetric. Exercise 3.3 Schwarz rotation symmetral) Let K K n with dim K = n, n 2. a K Kt) B n 1a) B n 1t) B n 1b) a Let rt) = t t b b l Without loss of generality we may consider the line given by l = { x 1, 0,..., 0) : x 1 R }, and we may also assume that K lies between the parallel hyperplanes {x R n : x 1 = a} and {x R n : x 1 = b}, for a < b. For each t [a, b], let Kt) = K {x R n : x 1 = t}, and we write B n 1 t) to denote the n 1)-dimensional ball with n 1)-dimensional volume vol n 1 Bn 1 t) ) ) = vol n 1 Kt) and centered in the point t, 0,..., 0). The Schwarz rotation symmetral of K is defined as K s = B n 1 t). t [a,b] vol n 1 Kt) ) /vol n 1 B n 1 )) 1/n 1) denote the radius of Bn 1 t). Show that rt) is a concave function in [a, b]. Exercise 3.4 Show that the Schwarz symmetral K s of a convex body K is also a convex body with vol K s ) = vol K). Discussion: 03.11.2008 List 3
Symmetrisations Exercise 2.1 Let H be a hyperplane and P be a polytope. Then st H P ) is a polytope. Exercise 2.2 A convex body is called unconditional if it is symmetric with respect to the coordinates hyperplanes. i) Show that each convex body can be transformed into an unconditional convex body via Steiner s symmetrisation. ii) Using Steiner s symmetrisation, establish the isodiametric inequality, i.e., 2 n vol K) vol B n )DK) n. Exercise 2.3 Let K = 1 2 K K) denote the central symmetrisation of K Kn. Then i) V 1 K) = V 1 K), ii) V i K) V i K) for i = 2,..., n, iii) DK) = DK), iv) rk) rk) and RK) RK). Exercise 2.4 For given K K n and u S n 1, the width of K in the direction u is defined as ωk, u) = hk, u)+hk, u). Then ωk) = min { ωk, u) : u S n 1} is called the minimal width of the convex body K. Show that i) ωk) = ωk); however ii) if H is a hyperplane, then the minimal width of st H K) can both increase or decrease. Discussion: 27.10.2008 List 2
Symmetrisations Exercise 1.1 Let K, K K n and let H be a hyperplane. Then: i) st H K K ) st H K) st H K ). ii) conv [ st H K) st H K ) ] st H [ conv K K ) ]. Show with examples that the reverse inclusions do not hold in general. Exercise 1.2 Let K i, K K n, i N, with dim K i = dim K = n, and let 0 int K. Then lim i K i = K if and only if, for any ε > 0, it holds 1 ε)k K i 1 + ε)k for any sufficiently large i. Exercise 1.3 Let H be a hyperplane. i) st H : {K K n : dim K = n} {K K n : dim K = n} is a continuous map. ii) Show with an example that the condition dim K = n can not be omitted; i.e., find a sequence K i K n, i N, of convex bodies converging to K K n with int K =, such that lim i st H K i ) st H K) strictly. Discussion: 20.10.2008 List 1