Exercise sheet 10. Discussion: Thursday,

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1 Exercise sheet 10 Discussion: Thursday, Exercise 10.1 Let K K n o, t > 0. Show that N (K, t B n ) N (K, 4t B n ) N (B n, (t/16)k ), N (B n, t K) N (B n, 4t K) N (K, (t/16)b n ). Hence, e.g., if B n 4K then N (B n, K) N (K, 1 16 B n). There is a theorem by Arstein-Milman-Szarek saying that there exists absolut constants α, β > 0 such that for all K K n o N (B n, α 1 K ) 1 β N (K, B n ) N (B n, αk ). Exercise 10.2 Let K K n o, r = (vol (K)/vol (B n )) 1/n and let N (K, rb n ) e c n for an absolute constant c. Then K is in M-position. Exercise 10.3 Let K K n be centered, i.e., the centroid is at the origin and let K K be in M-position with constant C. Then K ( K) is in M-position with constant c(c) depending only on C. Exercise 10.4 There exists an universal constant c 1 > 0 such that for K K n with 0 K we have vol (K)vol (K ) c n 1 vol (B n ) 2.

2 Exercise sheet 9 Discussion: Thursday, Exercise 9.1 Let v 1,..., v n R n and be a norm. Show that 2 n n ɛ i v i max{ v 1,..., v n }. ɛ { 1,1} n i=1 Exercise 9.2 Let K, L K n. For v int ((K L)/2) show hat is log-concave. ψ(v) = vol (( v + K) (v + L)) Exercise 9.3 Let K K n o and t R n. Show that γ n (t + K) e t 2 /2 γ n (K). Exercise 9.4 (Concentration of volume in convex bodies) Let K K n, vol (K) = 1, and L Ko n such that vol (K L) = r > 1/2. Then for t 1 ( ) 1 r (t+1)/2 vol (K (t L) c ) r, r where A c = R n \ A for a set A R n. First show that 2 t + 1 (t L)c + t 1 t + 1 L Lc. and then ask Brunn or Minkowski or both.

3 Exercise sheet 8 Discussion: Thursday, Exercise 8.1 Let v S n 1, t [0, 1] and let U = { u S n 1 : v, u t }. Then µ(u) 1 4e nt2 /4 and, in particular, for t 4/ n it holds µ(u) 0.9. Exercise 8.2 There exists always a δ-net on S n 1 consisting of at most (4/δ) n points. Exercise 8.3 For K K n o let M(K) = S n 1 u K dµ(v). i) Show that M(K) M(K ) 1. ii) Show that M(B p n) ( 2/π) n 1/p 1/2 for 1 p <. Rem.: For ii) Hölder and the next inequality might/could help. Exercise 8.4 Let κ n = vol (B n ) = π n/2 /Γ(n/2 + 1). Show that Show first that γ n = 2π n κ n 2π κ n 1 n + 1. κn κ n 1 is a decresing function in n and consider γ n γ n 1.

4 -7.teÜbung TU Berlin Exercise sheet 7 Discussion: Thursday, Exercise 7.1 Show that K n equipped with log d BM (, ) is a compact metric space, and that for K, L K n o d BM (K, L) = d BM (K, L ). Exercise 7.2 Let B p n = {x R n : n i=1 x i p 1} with B m = [ 1, 1] n. Show that for 1 p q 2 or 2 p q d BM (B p n, B q n) = n 1/p 1/q. The Minkowski-sum of finitely many line segments is called a zonotope Z, i.e., m Z = conv {v i, w i }. i=1 Exercise 7.3 Let Z be a zonotope. Then there exists a c R n and u i R n, 1 i m, such that m Z = c + conv { u i, u i }. For Z as above show that Exercise 7.4 vol (Z) = 2 n i=1 J [m], J =n det(u j : j J). i) Show that the projection body (see Exercise sheet 4) of a polytope is a zonotope. ii) What is the projection body of the cube [ 1/2, 1/2] n? iii) What is the projection body of an ellipsoid?

5 Exercise sheet 6 Discussion: Thursday, Exercise 6.1 Let K, L be 2-dimensional o-symmetric convex bodies. Show that the inequality vol (K) vol (L) is a consequence of either of the next two properties i) vol 1 (K lin {u}) vol 1 (L lin {u}) for all u S 1, ii) vol 1 (K lin {u} ) vol 1 (L lin {u} ) for all u S 1. Is symmetry needed? Exercise 6.2 Let K Kc n and let c 1, c 2 R >0 such that for all u S n 1 c 1 x, u 2 dx c 2. Show that c 1 L K c 2. K Exercise 6.3 Let K R n, L R m be two convex bodies in isotropic position, and let ( ) m ( ) n LL n+m LK n+m M = K L R n+m. L K L L Show that M is in isotropic position and L K L = L n n+m K L m n+m L. Exercise 6.4 Let a Z n, a 0, gcd(a) = 1 and let S(a) = { a, z : z N n 0 }. For x Rn let x 0 = #{x i 0 : 1 i n}. i) Let a < 2 n 1 and α S(a). Then there exists a z N n 0 with a, z = α and z 0 < n. ii) Let α S(a). Then there exists a z N n 0 with a, z = α and z 0 c log 2 a, where c is an absolute constant and a is the maximum norm. Exercise 4.4 could be helpful.

6 Exercise sheet 5 Discussion: Thursday, For K, L K n let N (K, L) = min{ S : S R n with K S + L} N (K, L) = min{ S : S K with K S + L} N (K, L) is called the covering number of K by L. For K, L K n, L = L let M (K, L) = max{ S : S K with x i x j L > 1 for all x i x j S}. M (K, L) is called the separation number of K by L. Exercise 5.1 Show that i) for K, L, M K n, K L: N (K, M) N (L, M), N (M, L) N (M, K), and N (M, L) N (M, K) ii) for K, L K n : N (K, L L) N (K, L) N (K, L). iii) for K K n, λ > 0: N (K, λ B n ) = N (K, λ B n ). iv) for K, L, M K n : N (K, L) N (K, M) N (M, L). Exercise 5.2 Let E 1, E 2 K n ellipsoids centered at the origin. Show that N (E 1, E 2 ) = N (E 2, E 1 ). Exercise 5.3 Let K, L K n. Show that N (K, (K K) L) = N (K, L). Exercise 5.4 Show that M (K, 2 L) N (K, L) N (K, L) M (K, L)

7 Exercise 5.5 Let K, L K n, dim K, dim L = n. Show that vol (K)/vol (L) N (K, L). If L = L then N (K, L) 2 n vol (K + L/2)/vol (L). Exercise 5.6 Let K K n c be in isotropic position. Then c 1 L K r(k) R(K) c 2 n L K, where c 1, c 2 > 0 are absolute constants.

8 Exercise sheet 4 Discussion: Thursday, Exercise 4.1 Show that among all n-simplices of inradius 1 the regular simplex has minimal volume. The inradius is the maximal radius of an n-dimensional ball contained in a body. Exercise 4.2 Show that n i=1 attains its minimum on the set {x R n 0 : n i=1 x i = α}, α > 0, if x 1 = x 2 = = x n. Exercise 4.3 Let K K n. The set Π(K) = { x R n : u, x vol n 1 (K u ), for all u S n 1} x x i i is called the projection body of K. Show that i) h ( Π(K), u ) = vol n 1 (K u ). ii) Π(A K) = det A A Π(K) for A GL(n, R), and Π(t + K) = Π(K) for t R n. Exercise 4.4 (Bombieri&Vaaler) Let a Z n, a 0, gcd(a) = 1. Show that there exists a z Z n \ {0} with a, z = 0 and max 1 j n z j a 1/(n 1). What could help is i) Minkowski s theorem, saying that every o-symmetric convex body with volume not less than 2 n det Λ contains a non-trivial lattice point of a lattice Λ, and ii) the set {z Z n : a, z = 0} is an (n 1)- dimensional lattice of determinant a.

9 Exercise 4.5 (McMullen) Let L be a k-dimensional linear subspace with orthogonal complement L. Show that vol k (C n L) = vol n k (C n L ). For a zonotope Z = m i=1 conv {0, v i} the volume is given by vol (Z) = det(v j : j J). J [m], J =n

10 Exercise sheet 3 Discussion: Thursday, !! A function f : M R n R >0 is called log-concave if ln f is a concave function. It is called centered if M x f(x) dx = 0. Exercise 3.1 Let K K n be a centered convex bdoy, i.e., K x dx = 0, and let L R n be a k-dimensional linear subspace with orthogonal subspace L. For y relint (K L) let f(y) = vol n k (K (y + L )). Show that f is a centered log-concave function. Exercise 3.2 Let F, G : R n R >0 be integrable log-concave functions. Then, their convolution (F G)(x) = F (x y) G(y) dy R n is also a log-concave function. The Prékopa-Leindler inequality might help. Exercise 3.3 Let K, L K n with 0 int K, int L, and let λ (0, 1). Show that i) ii) vol (K ) = 1 e h(k,x) dx. n! R n ln vol [λk + (1 λ)l] λ ln vol K + (1 λ) ln vol L. Here the Hölder inequality might help Exercise 3.4 Let Q R n 1 {0} be a polytope, y R n, y n 0, and P = conv {Q, y} be the pyramid over Q with apex y. Let c(p ) be the centroid of P, and let q Q be the intersection of Q with the ray y + λ(c(p ) y), λ 0. Then y c(p ) = n c(p ) q = n y q. n + 1

11 Exercise 3.5 i) Let P K n be an n-dimensional polytope, and let Q i P, i I, be polytopes forming a subdivision of P, i.e., i I Q i = P and dim(q i Q j ) n 1, i j. Find a relation between the centroids c(q i ) and c(p ). ii) Let S = conv {v 0,..., v n } be a n-dimensional simplex. c(p ) = 1 n n+1 i=0 v i. Show that Exercise 3.6 Let K K n mit c(k) = 0. i) For u R n let w(k, u) = h(k, u) + h(k, u). Show that ii) Show that K n K. 1 w(k, u) h(k, u) n n + 1 w(k, u). n + 1

12 Exercise sheet 2 Discussion: Thursday, Exercise 2.1 Let H(a, 0) be a 0 containing hyperplane and let K K n be symmetric with respect to H(a, 0). Let v R n \ {0} H(a, 0). Show that st H(v,0) (K) is still symmetric to H(a, 0). Exercise 2.2 Similar to the first part of the proof of the Rogers-Shephard inequality show that for K, L K n holds R n vol (K (x L)) dx = vol (K)vol (L), and conclude that that there exists a x 2 K such that vol (K (x K)) 2 n vol (K). Hence every convex body K of vol (K) = 2 n contains a centrally symmetric subset of volume at least 1. Exercise 2.3 Let K K n containing 0 in the interior and let h(k, ) be its support function. The function ρ(k, ) : R n \ {0} R 0 given by ρ(k, u) = max{ρ 0 : ρu K} is called radial function of K. Show that h(k, u)ρ(k, u) = 1. Exercise 2.4 Let K K n, K = K, dim K = n, and let L R n be a k-dimensional linear subspace with orthogonal complement L. Show that ( ) n 1 vol (K) k vol k (K L) vol n k (K L ) 1. For the lower bound it might be useful to observe that for x (K L ) a suitable translation of (1 ρ(k L, x) 1 )(K L) + x is contained in K (x + L).

13 Exercise 2.5 Cauchy s surface are formula states that for K K n F(K) = 1 vol n 1 (B n 1 ) S n 1 vol n 1 (K u ) du, K u denotes he orthogonal projection of K onto the hyperplane with normal u. Use it in order to show ( ) 1 F (K K) F (K). 2

14 Exercise sheet 1 Discussion: Thursday, Exercise 1.1 Let K, K i K n, i N, with 0 int K. Show that K i K if and only if for all ɛ > 0 there exists an i ɛ N such that for all i i ɛ (1 ɛ) K i K (1 + ɛ)k i. Exercise 1.2 Let K K n. Show that the Steiner-symmetral st H (K) of K with respect to a hyperplane H is a compact set. Exercise 1.3 Show that the minimal width of a convex boy may descrease or increase under Steiner-symemmetrizations. Exercise 1.4 Let K K n. Show that (without using Exercise 1.5) ( ) D(K) n vol (K) vol (B n ). 2 Exercise 1.5 Let K K n. The functional 1 w(k) = [h(k, u) + h(k, u)] du nvol (B n ) S n 1 is called the mean width of K. Here the integration du is meant with respect to the (n 1)-dimensional Hausdorff-measure. Let H be a hyperplane. Show that w(st H (K)) w(k), and conclude Urysohn s inequality ( ) w(k) n vol (K) vol (B n ). 2 Rem: The mean width is a continuous functional on K n. one more on the next page...

15 Exercise 1.6 Let T be an n-dimensional simplex, i.e., T = conv {v 0,..., v n }, v i R n, and dim T = n. i) Let H i,j, i j, be the hyperplane through 1 2 (v i + v j ) and with normal vector v i v j (orthogonal to the edge conv {v i, v j }). Show that st Hi,j (T ) is again an n-dimensional simplex. ii) Show that among all n-dimensional simplices T the ratio R(T )/r(t ) becomes minimal for a regular simplex. Here it might be helpful to use the fact the surface area of the Steiner symmetral st H (K) is strictly decreasing if K is not symmetric to the hypeprlane H.