On the monotonicity of perimeter of convex bodies

Transcription

1 On the monotonicity of perimeter of convex bodies Giorgio Stefani (SNS) Workshop on Calculus of Variations and Nonlinear PDEs Università degli Studi del Sannio - Benevento November 16-18, 2016 G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

2 Monotonicity of perimeter Let n 2 and A, B R n two convex bodies (compact, convex, interior ). A B = P(A) P(B) Well-known inequality, dates back to the ancient Greek (Archimedes postulated it in On the sphere and the cylinder). Several possible proofs: Cauchy formula for the area surface of convex bodies mixed volumes are monotone projection on convex closed sets is Lipschitz perimeter decreases under intersection with half-spaces Problem Lower bound on δ(b, A) = P(B) P(A) w.r.t. Hausdorff distance? G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

3 Lower bound for n = 2 First result: La Civita-Leonetti, 2008 (non-optimal bound) Theorem (Carozza-Giannetti-Leonetti-Passarelli di Napoli, 2015) Let A, B R 2 be two convex bodies. Then P(A) + 2h(A, B) 2 r 2 + h(a, B) 2 + r P(B), where L = {x R 2 : b a, x a = 0}, with a A and b B such that a b = h(a, B) and r = H1 (B L) 2. A a L h B b A a L r h B b (a) Setting of Theorem (b) Optimal configuration G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

4 Lower bound for n = 3 Theorem (Carozza-Giannetti-Leonetti-Passarelli di Napoli, 2016) Let A, B R 3 be two convex bodies. Then P(A) + πdh(a, B) 2 d 2 + h(a, B) 2 + d P(B), where H = {x R 3 : b a, x a 0}, with a A and b B such that a b = h(a, B) and d = dist(a, B H). H Note: distance d = dist(a, B H) replaces the bigger radius r = 2 (B H) π. B A a h H b (a) Setting of Theorem B A d a h b (b) Optimal configuration G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

5 Idea of the proof Since A B H and C B H c, we have B δ(b, A) = δ(b, B H) + δ(b H, A) δ(b, B H) A = H n 1 ( B H c ) H n 1 ( B H) H n 1 ( ) H n 1 (C base ) B H The problem reduces to min { H n 1 ( ) : given h and H n 1 (C base ) } C base a C b H ( ) Solution: n = 2: elementary calculus = isosceles triangle n = 3: parametrization of = right circular cone (d > 0 is necessary) Problem (Leonetti s dinner problem) Parametrization not easy for n 4. Other approach to solve ( ) for n 3? G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

6 Different approach We need to solve min { H n 1 ( ) : given h and H n 1 (C base ) } ( ) Schwartz symmetrization Given E R n convex body, define { ( H E Sch := x = (x, t) R n : x n 1 ) 1 } n 1 (E t ). Then P(E Sch ) P(E). ω n 1 Case E = C cone: slice to h = Hn 1 ((C Sch ) base ) = H n 1 (C base ) H n 1 ((C Sch ) lat ) H n 1 ( ) G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

7 Result Solution to ( ) for n 3 is a right circular cone with H n 1 (C base ) = ω n 1 r n 1 ( H n 1 ( B H) H n 1 ( ) = ω n 1 r n 2 r = h 2 + r 2 ω n 1 Theorem Let n 2. If A B are two convex bodies in R n, then H n 1 ( A) + ω n 1r n 2 h 2 h2 + r 2 + r Hn 1 ( B), where h = h(a, B) is the Hausdorff distance of A and B and H r = n 1 (B H) n 1, H = {x R n : b a, x a 0}, ω n 1 with a A and b B such that a b = h(a, B). Optimal configuration: right circular cone attached to a circular cylinder. ) 1 n 1 G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

8 Generalization: Wulff perimeter Let n 2 and let Φ: R n [0, ) be a positively 1-homogeneous convex function. Wulff Φ-perimeter E R n convex body, ν E inner normal = P Φ (E) = Φ(ν E ) dh n 1 Let A, B R n be two convex bodies: A B = P Φ (A) P Φ (B) Several possible proofs: Cauchy formula for the Wulff area surface of convex bodies mixed volumes are monotone Wulff perimeter decreases under intersection with half-spaces Problem Lower bound on δ Φ (B, A) = P Φ (B) P Φ (A) w.r.t. Hausdorff distance? E G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

9 Same strategy Since A B H and C B H c, we have ν H δ Φ (B, A) = δ Φ (B, B H) + δ Φ (B H, A) δ Φ (B, B H) = Φ(ν B ) dh n 1 Φ(ν H ) H n 1 ( B H) B H c Φ(ν C ) dh n 1 Φ(ν H ) H n 1 (C base ) B A B H C base a C b H The problem now reduces to { } min Φ(ν C ) dh n 1 : given h, ν H, and H n 1 (C base ) ( ) For P Φ translations and dilations are OK, rotations are not OK = ν H matters! Obstacle: no Schwartz symmetrization for P Φ for general Φ! G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

10 Admissible Φ Let n 2 and let Φ: R n [0, ) be a positively 1-homogeneous convex function. Definition We say that Φ is admissible if, for each ν S n 1, there exist g ν : [0, ) 2 [0, ), φ ν : ν [0, ), ν = span{ν} such that g ν 0 pos. 1-homog. and convex, s g ν (s, t) non-decreasing for each t; φ ν pos. 1-homog. and convex, coercive on ν (φ ν (z) > 0 z ν, z 0); ( it holds Φ(x) g ν φν (x (x ν)ν), x ν ) for x R n with x ν 0. Example 1: Φ coercive on R n (Φ ν (x) > 0, x 0) = Φ admissible, with φ ν (z) = z, g ν (s, t) = c s 2 + t 2, c = min x =1 Φ(x) Example 2: Φ(x) = x p := ( i x i p ) 1 p ν Sn 1 is coercive, but for ν = e n is better φ ν (z) = z p, g ν (s, t) = (s p + t p ) 1 p G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

11 Admissibility = Schwartz (from below) For Φ ν (x) = g ν (φ ν (x ), t) where x = x (x ν)ν ν, t = x ν R, we have Theorem (Van Schaftingen, 2006 & Baer, 2014) Given E R n convex body, define { E Sch,ν := x = (x, t) R n : x ( H n 1 ) 1 n 1 (E t ) H n 1 W φν }. (W φν ) where W φν ν is the φ ν -Wulff shape in ν. Then P Φν (E Sch,ν ) P Φν (E). = we could use this general Schwartz symm. to estimate ( ) from below: Φ(ν C ) dh n 1 Φadmiss. Φ νh (ν C ) dh n 1 ( ) H n 1 (W φνh )r n 2 g νh (h, r) where r = n 1 H n 1 (B H) H n 1 (W νh ) and ( ) = Schwartz + coarea, subdifferential,... G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

12 General result We get δ Φ (B, A) Φ(ν C ) dh n 1 Φ(ν H ) H n 1 (C base ) H n 1 (W φνh )r n 2 g νh (h, r) Φ(ν H ) H n 1 (W φνh )r n 1 Theorem Let n 2 and Φ: R n [0, ) pos. 1-homog., convex, admissible. If A B are two convex bodies in R n, then P Φ (A) + H n 1 (W νh )r n 2( g νh (h, r) Φ(ν H )r ) + PΦ (B), where h = h(a, B) is the Hausdorff distance of A and B and H r = n 1 (B H) n 1 H n 1, H = {x R n : b a, x a 0}, ν H = a b (W νh ) a b, with a A and b B such that a b = h(a, B). G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

13 Remarks Remark 1. In general Φ > Φ νh, so the result is not optimal! When Φ = Φ νh, the optimal configuration is B right cone with base W νh + cylinder with base W νh ν H A W νh Remark 2. Too much technology for cones! (Baer s setting: finite perimeter sets) Can we find a simpler approach? (avoid Schwartz, coarea, subdifferential...) YES, we can follow the following scheme: Assume C base is a polytope and reduce the problem to ν H. Apply Wulff inequality in νh to prove Φ(ν C ) dh n 1 H n 1 (W φνh )r n 2 g νh (h, r) Extend to any C base by approximation. G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14

14 Thank you for your attention! G. Stefani (SNS) On the monotonicity of perimeter November 16-18, / 14