Hyperplane sections of convex bodies

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1 Hyperplane sections of convex bodies Carla Peri Università Cattolica S.C. - Milano Fourth International Workshop on Convex Geometry - Analytic Aspects Cortona, June 4th - 8th, 2007

2 K R n convex body

3 K R n convex body u S n 1 u o

4 K R n convex body u S n 1, canonical scalar product u := {x R n : x, u = 0} u o u

5 K R n convex body u S n 1, canonical scalar product u := {x R n : x, u = 0} H u,t := {x R n : x, u = t}, t R H u,t t u o u

6 K R n convex body u S n 1, canonical scalar product u := {x R n : x, u = 0} H u,t := {x R n : x, u = t}, t R A : volume of A R n (relative to its affine convex hull) t H u,t u o u

7 H u,t + := {x R n : x, u t}, Hu,t := {x R n : x, u t}, t R Problem. Find the best upper bound for the ratio ρ K,u (t) := min{ K H u,t +, K Hu,t }. K H u,t

8 H + u,t := {x R n : x, u t}, H u,t := {x R n : x, u t}, t R Problem. Find the best upper bound for the ratio ρ K,u (t) := min{ K H u,t +, K Hu,t }. K H u,t C. P. (1999): ρ K,u (t) 2 ( ) 1 ln 2 min x y, u dx y R n

9 H + u,t := {x R n : x, u t}, H u,t := {x R n : x, u t}, t R Problem. Find the best upper bound for the ratio ρ K,u (t) := min{ K H u,t +, K Hu,t }. K H u,t C. P. (1999): ρ K,u (t) 2 ( ) 1 ln 2 min x y, u dx y R n If K = K, then ρ K,u (t) 2 x, u dx

10 These type of inequalities are simpler versions of the so-called relative isoperimetric inequalities, which, for every sufficiently smooth subset E of a domain G, bound from above the Lebesgue measure either of E or G\E by an appropriate (n 1)-dimensional measure (P(E, G)) of the boundary E G.

11 These type of inequalities are simpler versions of the so-called relative isoperimetric inequalities, which, for every sufficiently smooth subset E of a domain G, bound from above the Lebesgue measure either of E or G\E by an appropriate (n 1)-dimensional measure (P(E, G)) of the boundary E G. M. Dyer e A. Frieze (1992): K R n convex body min( E, K \ E ) P(E, K) D 2 (D: diameter of K)

12 These type of inequalities are simpler versions of the so-called relative isoperimetric inequalities, which, for every sufficiently smooth subset E of a domain G, bound from above the Lebesgue measure either of E or G\E by an appropriate (n 1)-dimensional measure (P(E, G)) of the boundary E G. M. Dyer e A. Frieze (1992): K R n convex body min( E, K \ E ) P(E, K) D 2 (D: diameter of K) R. Kannan, L. Lovász e M. Simonovits (1995): K R n with centroid at the origin min( E, K \ E ) P(E, K) 2 ( ) 1 x dx ln 2

13 Proposition. The maximum of the ratio ρ K,u (t), over the set of parallel hyperplane sections of K, is attained when the section bisects the volume of K: ρ K,u (t) = min{ K H u,t +, K Hu,t } K H u,t K 2 K H u, (H u : hyperplane orthogonal to u which cuts K into two parts with equal volumes.)

14 D. Hensley (1980): K = K, u S n 1 K 2 K u ( ) x, u 2 2 dx

15 D. Hensley (1980): K = K, u S n 1 K 2 K u ( ) x, u 2 2 dx V.D. Milman and A. Pajor (1989): K = K, p > 0 ( ) 1 1 c 2 x, u p p dx ( ) 1 K 1 2 K u c 1 x, u p p dx

16 D. Hensley (1980): K = K, u S n 1 K 2 K u ( ) x, u 2 2 dx V.D. Milman and A. Pajor (1989): K = K, p > 0 ( ) 1 1 c 2 x, u p p dx ( ) 1 K 1 2 K u c 1 x, u p p dx c 1 := c 1 (p) = (p + 1) 1/p, c 2 := c 2 (p, n) = 1 n ( ) 1/p (p+1)(p+2)...(p+n) n!

17 D. Hensley (1980): K = K, u S n 1 K 2 K u ( ) x, u 2 2 dx V.D. Milman and A. Pajor (1989): K = K, p > 0 ( ) 1 1 c 2 x, u p p dx ( ) 1 K 1 2 K u c 1 x, u p p dx c 1 := c 1 (p) = (p + 1) 1/p, c 2 := c 2 (p, n) = 1 n ( (p+1)(p+2)...(p+n) n! ) 1/p (c 2 (p, n) Γ(p + 1) 1/p when n )

18 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia

19 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia M. Fradelizi (1999): K with the centroid at the origin, p 1 c 2 I p (K, u) K 2 K u c 1I p (K, u) (1)

20 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia M. Fradelizi (1999): K with the centroid at the origin, p 1 c 2 I p (K, u) K 2 K u c 1I p (K, u) (1) There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a double-cone in the direction u.

21 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia M. Fradelizi (1999): K with the centroid at the origin, p 1 c 2 I p (K, u) K 2 K u c 1I p (K, u) (1) There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a double-cone in the direction u. Extension: φ : R R even convex function, α = K 2 K u

22 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia M. Fradelizi (1999): K with the centroid at the origin, p 1 c 2 I p (K, u) K 2 K u c 1I p (K, u) (1) There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a double-cone in the direction u. Extension: φ : R R even convex function, α = 1 1 φ(αt)dt 2 n φ( x, u )dx φ(αt) n K 2 K u ( 1 t n ) n 1 dt (2)

23 ( ) 1 1 I p (K, u) := x, u p p dx p-th moment of inerzia M. Fradelizi (1999): K with the centroid at the origin, p 1 c 2 I p (K, u) K 2 K u c 1I p (K, u) (1) There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a double-cone in the direction u. Extension: φ : R R even convex function, α = 1 1 φ(αt)dt 2 n φ( x, u )dx φ(αt) n K 2 K u ( 1 t n ) n 1 dt (2) with φ(t) := t p, p 1. (2) (1)

24 c 3 I p (K, u) K 2 max t R K H u,t c 1I p (K, u)

25 c 3 I p (K, u) K 2 max t R K H u,t c 1I p (K, u) c 3 := c 3 (p, n) = 1 2 ( ( ) n n+p n ( ) t p 1 t ) 1 n 1 p dt n n

26 c 3 I p (K, u) K 2 max t R K H u,t c 1I p (K, u) c 3 := c 3 (p, n) = 1 2 ( ( ) n n+p n ( ) t p 1 t ) 1 n 1 p dt n n There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a cone in the direction u.

27 c 3 I p (K, u) K 2 max t R K H u,t c 1I p (K, u) c 3 := c 3 (p, n) = 1 2 ( ( ) n n+p n ( ) t p 1 t ) 1 n 1 p dt n n There is equality in the right hand side if and only if K is a cylinder in the direction u, and in the left hand side if and only if K is a cone in the direction u. c 3 (p, n) 1 2 ( ) t 1 p e t p dt when n

28 Extension: φ : R R even convex function, γ = K 2 max t R K H u,t 1 1 φ(γt)dt 2 φ( x, u )dx ( 2 n (1+ n) 1 n 1 φ 2nγt n+1 ) (1 t n) n 1 dt

29 Extension: φ : R R even convex function, γ = K 2 max t R K H u,t 1 1 φ(γt)dt 2 φ( x, u )dx ( 2 n (1+ n) 1 n 1 φ 2nγt n+1 ) (1 t n) n 1 dt Equality cases are characterized as above when φ is strictly convex.

30 Extension: φ : R R even convex function, γ = K 2 max t R K H u,t 1 1 φ(γt)dt 2 φ( x, u )dx ( 2 n (1+ n) 1 n 1 φ 2nγt n+1 ) (1 t n) n 1 dt Equality cases are characterized as above when φ is strictly convex. C. Schütt (1997): Applications to the floating body and the illumination body. M. Meyer e E. Werner (1998): Applications to Santaló regions.

31 Problem. Find an upper bound for the relative isoperimetric ratio ρ K,u (t) in terms of the p-th moments of inertia of K. Theorem 1. Let K be a convex body in R n whose centroid is at the origin. Let φ : R R be an even convex function. Let u S n 1 and α = K /(2 K H u ), where H u denotes the hyperplane orthogonal to u which bisects the volume of K. Then 1 1 φ(αt)dt 1 φ( x, u )dx (3) 2 1 The inequality is sharp: if φ is strictly convex, there is equality if and only if K is a cylinder in the direction u.

32 Corollary 1.. Let K be a convex body in R n whose centroid is at the origin. Let u S n 1, and p 1. Denote by H u the hyperplane orthogonal to u which bisects the volume of K. Then ( ) 1 K p K H u x, u p p dx. There is equality if and only if K is a cylinder in the direction u

33 Theorem 2. Let K be a convex body in R n whose centroid is at the origin, and p 1. Let H u,t be a hyperplane orthogonal to u S n 1, which cuts K into two parts, say K H + u,t and K H u,t. Then min { K H u,t +, K Hu,t } H u,t K ( p + 1 K K ) 1 x, u p p dx. Equality holds if K is a cylinder in the direction u.

34 Theorem 2. Let K be a convex body in R n whose centroid is at the origin, and p 1. Let H u,t be a hyperplane orthogonal to u S n 1, which cuts K into two parts, say K H + u,t and K H u,t. Then min { K H u,t +, K Hu,t } H u,t K ( p + 1 K K ) 1 x, u p p dx. Equality holds if K is a cylinder in the direction u. Corollary 2. Let K be in isotropic position, and K = 1. Then min { K H u,t +, K Hu,t } 3L K < cn 1 4. H u,t K (L K : isotropic constant of K.)

35 Outline of the proof of Theorem 1. Functional version of (3): Let f (t) := K H u,t and f ( t) := K H u = 1, then + + K = f (t)dt α = f (t)dt t = f (t)dt 2f ( t) K φ( x, u )dx = + φ(t)f (t)dt

36 Outline of the proof of Theorem 1. Functional version of (3): Let f (t) := K H u,t and f ( t) := K H u = 1, then + + K = f (t)dt α = f (t)dt t = f (t)dt 2f ( t) We have to prove K φ( x, u )dx = α α φ(t)dt for every n 1-concave integrable function φ(t)f (t)dt φ(t)f (t)dt + tf (t)dt = 0, t f (t)dt = + t f (t)dt

37 Step 1. Reduction to the log-affine case There exists a log-affine function g(t) = e a(t t) χ [ c,b] (t), where [ c, b] support (f ), such that t g(t)dt = t f (t)dt, + t g(t)dt = + t f (t)dt, + + φ(t)g(t)dt tg(t)dt = 0 for every convex function φ : R R. + φ(t)f (t)dt (4)

38 Lemma. ( Fradelizi, 1999) Let v : R R be an integrable compactly supported function such that + v(t)dt = 0 and + tv(t)dt = 0. Set V(t) = t v(s)ds. Let φ be a convex function and µ be the positive Borel measure on R such that φ = µ. Then the function W(t) = t V(s)ds is compactly supported and + φ(t)v(t)dt = + W(t)dµ(t). (v := f g) Inequality (4) follows from W 0.

39 Step 2. Reduction to the constant case. Among all log-affine functions the quantity + φ(t)g(t)dt is minimal when g is constant on its support.

40 Step 2. Reduction to the constant case. Among all log-affine functions the quantity + φ(t)g(t)dt is minimal when g is constant on its support. Set d := g(t)dt. Then we have to prove that d d φ(t)dt b c φ(t)g(t)dt, (5) where g(t) = e (t t) χ [ c,b] (t)

41 If we set x := (b + c)/2, then b, c, d are C positive functions of x > 0. Inequality (5) becomes I(x) := b(x) c(x) d(x) φ(t)e (t t) dt φ(t)dt 0 d(x)

42 If we set x := (b + c)/2, then b, c, d are C positive functions of x > 0. Inequality (5) becomes b(x) d(x) I(x) := φ(t)e (t t) dt φ(t)dt 0 c(x) d(x) We have I(0) = 0, and by differentiating we get b(x) c(x) te (t t) dt = 0 d(x) = 1 2 b(x) c(x) e (t t) dt

43 If we set x := (b + c)/2, then b, c, d are C positive functions of x > 0. Inequality (5) becomes b(x) d(x) I(x) := φ(t)e (t t) dt φ(t)dt 0 c(x) d(x) We have I(0) = 0, and by differentiating b(x) c(x) te (t t) dt = 0 d(x) = 1 2 b(x) c(x) e (t t) dt we get ( )( b + c c I = b e (b t) c b + c φ(b) + b ) φ(c) φ(d) b + c ( )( ( ) ) b + c 2bc b e (b t) φ φ(d) 0 c b + c I(x) 0

44 The equality case. There is equality if and only if there is equality in the corresponding functional form, that is + φ(t)g(t)dt = + φ(t)f (t)dt where g(t) = f ( t)χ [ α,α] (t).

45 The equality case. There is equality if and only if there is equality in the corresponding functional form, that is + φ(t)g(t)dt = + φ(t)f (t)dt where g(t) = f ( t)χ [ α,α] (t). Then by the previous lemma the second primitive W of v = f g satisfies + W(t)dµ(t) = 0 where W 0. φ strictly convex µ positive W = 0 f = g

Hyperplane Sections of Convex Bodies in Isotropic Position

Hyperplane Sections of Convex Bodies in Isotropic Position Beiträge zur Algebra ud Geometrie Cotributios to Algebra ad Geometry Volume 4 999), No., 63-83. Hyperplae Sectios of Covex Bodies i Isotropic Positio Matthieu Fradelizi Uiversité de Mare-la-Vallée, Equipe