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Chiara Bianchini & Andrea Colesanti Università degli Studi di Firenze A sharp Rogers-Shephard type inequality for the p-difference body of a planar convex body “Affine Invariants, Randomness, and Approximation in Convex Geometry” Hoboken, April 14-15, 2007 The difference body of a convex body The difference body of a convex body ◮ Let K be a convex body in Rn ; the difference body of K is DK := K + (−K ) = {x + y | x ∈ K , −y ∈ K } . DK is a centrally symmetric convex body. The difference body of a convex body ◮ Let K be a convex body in Rn ; the difference body of K is DK := K + (−K ) = {x + y | x ∈ K , −y ∈ K } . DK is a centrally symmetric convex body. ◮ A straightforward application of the Brunn-Minkowski inequality gives Vn (DK ) ≥ 2n Vn (K ) for every convex body K (here Vn denotes the n-dimensional volume). The Rogers-Shephard inequality The Rogers-Shephard inequality Theorem (Rogers & Shephard, 1957). For every convex body K 2n Vn (DK ) ≤ Vn (K ) . n The Rogers-Shephard inequality Theorem (Rogers & Shephard, 1957). For every convex body K 2n Vn (DK ) ≤ Vn (K ) . n Equality holds if and only if K is a simplex. The Rogers-Shephard inequality Theorem (Rogers & Shephard, 1957). For every convex body K 2n Vn (DK ) ≤ Vn (K ) . n Equality holds if and only if K is a simplex. Theorem (Rogers & Shephard, 1958). Let K be a convex body containing the origin and let conv(K ∪ (−K )) be the convex hull of K ∪ (−K ). The Rogers-Shephard inequality Theorem (Rogers & Shephard, 1957). For every convex body K 2n Vn (DK ) ≤ Vn (K ) . n Equality holds if and only if K is a simplex. Theorem (Rogers & Shephard, 1958). Let K be a convex body containing the origin and let conv(K ∪ (−K )) be the convex hull of K ∪ (−K ).Then Vn (conv(K ∪ (−K ))) ≤ 2n Vn (K ) . The Rogers-Shephard inequality Theorem (Rogers & Shephard, 1957). For every convex body K 2n Vn (DK ) ≤ Vn (K ) . n Equality holds if and only if K is a simplex. Theorem (Rogers & Shephard, 1958). Let K be a convex body containing the origin and let conv(K ∪ (−K )) be the convex hull of K ∪ (−K ).Then Vn (conv(K ∪ (−K ))) ≤ 2n Vn (K ) . Equality holds if and only if K is a simplex with one vertex at the origin. The p-sum of convex bodies (Firey, 1962) The p-sum of convex bodies (Firey, 1962) Let p ≥ 1. Let K and L be convex bodies in Rn , containing the origin, and denote by hK and hL the corresponding support functions. The p-sum of convex bodies (Firey, 1962) Let p ≥ 1. Let K and L be convex bodies in Rn , containing the origin, and denote by hK and hL the corresponding support functions. The p-sum of K and L K +p L is the convex body having as support function p hK +p L = (hK + hLp )1/p . The p-sum of convex bodies (Firey, 1962) Let p ≥ 1. Let K and L be convex bodies in Rn , containing the origin, and denote by hK and hL the corresponding support functions. The p-sum of K and L K +p L is the convex body having as support function p hK +p L = (hK + hLp )1/p . For p = ∞ set hK +∞ L = max{hK , hL } . The p-sum of convex bodies (Firey, 1962) Let p ≥ 1. Let K and L be convex bodies in Rn , containing the origin, and denote by hK and hL the corresponding support functions. The p-sum of K and L K +p L is the convex body having as support function p hK +p L = (hK + hLp )1/p . For p = ∞ set hK +∞ L = max{hK , hL } . Note that K +1 L = K + L and K +∞ L = conv(K ∪ L) . The p-difference body The p-difference body Let K be a convex body in Rn containing the origin and let p ∈ [1, ∞]. The p-difference body Let K be a convex body in Rn containing the origin and let p ∈ [1, ∞]. The p-difference body of K is Dp K = K +p (−K ) . The p-difference body Let K be a convex body in Rn containing the origin and let p ∈ [1, ∞]. The p-difference body of K is Dp K = K +p (−K ) . ◮ For p = 1: D1 K = DK and Vn (D1 K ) ≤ 2n Vn (K ) . n The p-difference body Let K be a convex body in Rn containing the origin and let p ∈ [1, ∞]. The p-difference body of K is Dp K = K +p (−K ) . ◮ For p = 1: D1 K = DK and Vn (D1 K ) ≤ ◮ 2n Vn (K ) . n For p = ∞: D∞ K = conv(K ∪ (−K )) and Vn (D∞ K ) ≤ 2n Vn (K ) . The problem The problem Given p > 1, find the optimal constant c = c(n, p) such that for every convex body K in Rn containing the origin Vn (Dp K ) ≤ c(n, p)Vn (K ) . The problem Given p > 1, find the optimal constant c = c(n, p) such that for every convex body K in Rn containing the origin Vn (Dp K ) ≤ c(n, p)Vn (K ) . Remark. For every p and K Dp K ⊂ DK and Dp K ⊂ 21/p D∞ K . The problem Given p > 1, find the optimal constant c = c(n, p) such that for every convex body K in Rn containing the origin Vn (Dp K ) ≤ c(n, p)Vn (K ) . Remark. For every p and K Dp K ⊂ DK and Dp K ⊂ 21/p D∞ K . Consequently Vn (Dp K ) ≤ min n(1+p) 2n p ,2 Vn (K ) . n The two-dimensional case The two-dimensional case Theorem (C. Bianchini & A. C., 2006). For every planar convex body K containing the origin and for every p > 1 we have V2 (Dp K ) ≤ c(2, p)V2 (K ) , where The two-dimensional case Theorem (C. Bianchini & A. C., 2006). For every planar convex body K containing the origin and for every p > 1 we have V2 (Dp K ) ≤ c(2, p)V2 (K ) , where c(2, p) = 2 + 2(p − 1) Z π 2 0 sinp−2 t cosp−2 t 2 (p−1) dt . p p sin t + cosp t The two-dimensional case Theorem (C. Bianchini & A. C., 2006). For every planar convex body K containing the origin and for every p > 1 we have V2 (Dp K ) ≤ c(2, p)V2 (K ) , where c(2, p) = 2 + 2(p − 1) Z π 2 0 sinp−2 t cosp−2 t 2 (p−1) dt . p p sin t + cosp t Equality holds if K is a triangle with one vertex at the origin. The two-dimensional case Theorem (C. Bianchini & A. C., 2006). For every planar convex body K containing the origin and for every p > 1 we have V2 (Dp K ) ≤ c(2, p)V2 (K ) , where c(2, p) = 2 + 2(p − 1) Z π 2 0 sinp−2 t cosp−2 t 2 (p−1) dt . p p sin t + cosp t Equality holds if K is a triangle with one vertex at the origin. Linear parameter systems Linear parameter systems A linear parameter system is a family of convex bodies {Kt } that can be written in the form Kt = conv{xi + λi tv : i ∈ I }, t ∈ [a, b] ; Linear parameter systems A linear parameter system is a family of convex bodies {Kt } that can be written in the form Kt = conv{xi + λi tv : i ∈ I }, where: I is an arbitrary index set, t ∈ [a, b] ; Linear parameter systems A linear parameter system is a family of convex bodies {Kt } that can be written in the form Kt = conv{xi + λi tv : i ∈ I }, t ∈ [a, b] ; where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded subsets of Rn and of R respectively Linear parameter systems A linear parameter system is a family of convex bodies {Kt } that can be written in the form Kt = conv{xi + λi tv : i ∈ I }, t ∈ [a, b] ; where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded subsets of Rn and of R respectively and v ∈ Rn is the direction of the linear parameter system. Linear parameter systems A linear parameter system is a family of convex bodies {Kt } that can be written in the form Kt = conv{xi + λi tv : i ∈ I }, t ∈ [a, b] ; where: I is an arbitrary index set, {xi }i ∈I and {λi }i ∈I are bounded subsets of Rn and of R respectively and v ∈ Rn is the direction of the linear parameter system. Kt Kt 0 v 1 Kt 2 Convexity of the volume Convexity of the volume Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a linear parameter system is a convex function of t. Convexity of the volume Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a linear parameter system is a convex function of t. ◮ Campi, C. & Gronchi, 1999 (Sylvester’s problem). Convexity of the volume Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a linear parameter system is a convex function of t. ◮ Campi, C. & Gronchi, 1999 (Sylvester’s problem). ◮ Campi & Gronchi, 2002 (Lp -centroid inequality). Convexity of the volume Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a linear parameter system is a convex function of t. ◮ Campi, C. & Gronchi, 1999 (Sylvester’s problem). ◮ Campi & Gronchi, 2002 (Lp -centroid inequality). ◮ Campi & Gronchi, 2006 (volume product). Convexity of the volume Theorem (Rogers & Shephard, 1958). The volume Vn (Kt ) of a linear parameter system is a convex function of t. ◮ Campi, C. & Gronchi, 1999 (Sylvester’s problem). ◮ Campi & Gronchi, 2002 (Lp -centroid inequality). ◮ Campi & Gronchi, 2006 (volume product). ◮ Meyer & Reisner, 2006 (volume product). p-sums of linear parameter systems p-sums of linear parameter systems Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and {Lt }t∈[a,b] be linear parameter systems along the same direction s.t. 0 ∈ Kt , Lt for every t, then {Kt +p Lt }t∈[a,b] is also a linear parameter system. p-sums of linear parameter systems Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and {Lt }t∈[a,b] be linear parameter systems along the same direction s.t. 0 ∈ Kt , Lt for every t, then {Kt +p Lt }t∈[a,b] is also a linear parameter system. p = 1: Campi & Gronchi, 2006. p-sums of linear parameter systems Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and {Lt }t∈[a,b] be linear parameter systems along the same direction s.t. 0 ∈ Kt , Lt for every t, then {Kt +p Lt }t∈[a,b] is also a linear parameter system. p = 1: Campi & Gronchi, 2006. Corollary. If {Kt }t∈[a,b] is a linear parameter system s.t. 0 ∈ Kt for every t, then {Dp Kt }t∈[a,b] is also a linear parameter system. p-sums of linear parameter systems Theorem (C. Bianchini & A. C., 2006). Let {Kt }t∈[a,b] and {Lt }t∈[a,b] be linear parameter systems along the same direction s.t. 0 ∈ Kt , Lt for every t, then {Kt +p Lt }t∈[a,b] is also a linear parameter system. p = 1: Campi & Gronchi, 2006. Corollary. If {Kt }t∈[a,b] is a linear parameter system s.t. 0 ∈ Kt for every t, then {Dp Kt }t∈[a,b] is also a linear parameter system. In particular Vn (Dp Kt ) is a convex function of t. Parallel chord movements Parallel chord movements These are particular linear parameter systems: Kt = {x + α(x)tv : x ∈ K }, t ∈ [a, b], where the functions α is constant on each chord parallel to v and such that convexity is preserved for every t. Parallel chord movements These are particular linear parameter systems: Kt = {x + α(x)tv : x ∈ K }, t ∈ [a, b], where the functions α is constant on each chord parallel to v and such that convexity is preserved for every t. K Kt v Parallel chord movements These are particular linear parameter systems: Kt = {x + α(x)tv : x ∈ K }, t ∈ [a, b], where the functions α is constant on each chord parallel to v and such that convexity is preserved for every t. K Kt v Remark. The volume is constant along a parallel chord movement. The first main ingredient of the proof The first main ingredient of the proof Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex bodies s.t. 0 ∈ Kt for every t, then Vn (Dp Kt ) , Vn (Kt ) is a convex function of t. t ∈ [a, b] , The first main ingredient of the proof Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex bodies s.t. 0 ∈ Kt for every t, then Vn (Dp Kt ) , Vn (Kt ) t ∈ [a, b] , is a convex function of t. In particular the above quotient attains its maximum either for t = a or for t = b. The first main ingredient of the proof Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex bodies s.t. 0 ∈ Kt for every t, then Vn (Dp Kt ) , Vn (Kt ) t ∈ [a, b] , is a convex function of t. In particular the above quotient attains its maximum either for t = a or for t = b. Proof. ◮ {Dp Kt }t∈[a,b] is a linear parameter system and then Vn (Dp Kt ) is convex w.r.t. t. The first main ingredient of the proof Theorem. If {Kt }t∈[a,b] is a parallel chord movement of convex bodies s.t. 0 ∈ Kt for every t, then Vn (Dp Kt ) , Vn (Kt ) t ∈ [a, b] , is a convex function of t. In particular the above quotient attains its maximum either for t = a or for t = b. Proof. ◮ {Dp Kt }t∈[a,b] is a linear parameter system and then Vn (Dp Kt ) is convex w.r.t. t. ◮ Vn (Kt ) is constant w.r.t. t. Polygons Polygons For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for every t and ◮ P ∈ {Pt }t∈[a,b] ; Polygons For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for every t and ◮ P ∈ {Pt }t∈[a,b] ; ◮ Pa and Pb are both polygons having at most (k − 1) vertices. Polygons For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for every t and ◮ P ∈ {Pt }t∈[a,b] ; ◮ Pa and Pb are both polygons having at most (k − 1) vertices. v v P Polygons For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a parallel chord movement {Pt }t∈[a,b] s.t. 0 ∈ Pt for every t and ◮ P ∈ {Pt }t∈[a,b] ; ◮ Pa and Pb are both polygons having at most (k − 1) vertices. v v P V2 (Dp P) ≤ max V2 (P) V2 (Dp Pa ) V2 (Dp Pb ) , V2 (Pa ) V2 (Pb ) From polygons to triangles From polygons to triangles ◮ For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a polygon P ′ with at most (k − 1) vertices and s.t. 0 ∈ P ′ and V2 (Dp P) V2 (Dp P ′ ) ≤ . V2 (P) V2 (P ′ ) From polygons to triangles ◮ For every convex polygon P with k vertices, k > 3, and s.t. 0 ∈ P, there exists a polygon P ′ with at most (k − 1) vertices and s.t. 0 ∈ P ′ and V2 (Dp P) V2 (Dp P ′ ) ≤ . V2 (P) V2 (P ′ ) ◮ For every convex polygon P s.t. 0 ∈ P there exists a triangle T s.t. 0 ∈ T and V2 (Dp T ) V2 (Dp P) ≤ . V2 (P) V2 (T ) Putting a vertex in the origin in two moves Putting a vertex in the origin in two moves Let T ′ be a triangle containing 0. There exists a triangle T with one vertex at the origin such that V2 (Dp T ′ ) V2 (Dp T ) ≤ . V2 (T ′ ) V2 (T ) Putting a vertex in the origin in two moves Let T ′ be a triangle containing 0. There exists a triangle T with one vertex at the origin such that V2 (Dp T ′ ) V2 (Dp T ) ≤ . V2 (T ′ ) V2 (T ) To prove it, we use the fact that translations are parallel chord movements. Putting a vertex in the origin in two moves Let T ′ be a triangle containing 0. There exists a triangle T with one vertex at the origin such that V2 (Dp T ′ ) V2 (Dp T ) ≤ . V2 (T ′ ) V2 (T ) To prove it, we use the fact that translations are parallel chord movements. T’ v o o v o Putting a vertex in the origin in two moves Let T ′ be a triangle containing 0. There exists a triangle T with one vertex at the origin such that V2 (Dp T ′ ) V2 (Dp T ) ≤ . V2 (T ′ ) V2 (T ) To prove it, we use the fact that translations are parallel chord movements. T’ v o v o o v o v o o T’ The computation of c(2, p) The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . c(2, p) = sup V2 (Dp K ) : K = planar convex body, 0 ∈ K V2 (K ) The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . V2 (Dp K ) : K = planar convex body, 0 ∈ K V2 (K ) V2 (Dp P) : P = polygon, 0 ∈ T = sup V2 (P) c(2, p) = sup The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . V2 (Dp K ) : K = planar convex body, 0 ∈ K V2 (K ) V2 (Dp P) : P = polygon, 0 ∈ T = sup V2 (P) V2 (Dp T ) : T = triangle, 0 ∈ T = sup V2 (T ) c(2, p) = sup The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . V2 (Dp K ) V2 (K ) V2 (Dp P) = sup V2 (P) V2 (Dp T ) = sup V2 (T ) V2 (Dp T ) = sup V2 (T ) c(2, p) = sup : K = planar convex body, 0 ∈ K : P = polygon, 0 ∈ T : T = triangle, 0 ∈ T : T = triangle, 0 is a vertex of T The computation of c(2, p) c(2, p) is the optimal constant such that for every planar convex body K containing the origin we have V2 (Dp K ) ≤ c(2, p)V2 (K ) . = = = = V2 (Dp K ) V2 (K ) V2 (Dp P) sup V2 (P) V2 (Dp T ) sup V2 (T ) V2 (Dp T ) sup V2 (T ) V2 (Dp T ) , T V2 (T ) c(2, p) = sup : K = planar convex body, 0 ∈ K : P = polygon, 0 ∈ T : T = triangle, 0 ∈ T : T = triangle, 0 is a vertex of T = any triangle with a vertex at 0.