II

II 2009 1

II Euclid Euclid

i 1 1 2 7 3 11 4 18 5 20 6 22 7 26 8 Hausdorff 29 36

1 1 1.1 E n n Euclid E n n R n = {(x 1,..., x n ) x i R} x, y = x 1 y 1 + + x n y n (x, y E n ), x = x, x 1/2 = { (x 1 ) 2 + + (x n ) 2} 1/2 (x E n ) 1.2 x 1,..., x k E n λ 1,..., λ k R λ 1 x 1 + + λ k x k x 1,..., x k A E n A lina A lina A lina { } lina = λ i x i λ i R, x i A k lina ={λx λ R, x A} {λ 1 x 1 + λ 2 x 2 λ 1, λ 2 R, x 1, x 2 A} { 3 } λ i x i λ i R, x i A = k N { } λ i x i λ i R, x i A. { } λ i x i λ i R, x i A k=1 k λ i x i A lina

2 1. 1.3 x 1,..., x k E n λ 1 x 1 + + λ k x k = 0, λ i R λ 1 = = λ k = 0 x 1,..., x k 1.4 E n V x, y V x + y V, α R, x V αx V E n V x 1,..., x k lin{x 1,..., x k } = V x 1,..., x k V k V dim V = k V 1.5 V E m W E n f : V W f(x + y) = f(x) + f(y) (x, y E m ), f(αx) = αf(x) (α R, x R). 1.6 x 1,..., x k E n λ 1 + + λ k = 1 λ 1,..., λ k R λ 1 x 1 + + λ k x k x 1,..., x k A E n A affa A 0 0 lin{0} = {0} 0 x E n llin{x} 1 x E n aff{x} 1 x {x} x 0 x 1, x 2 E n lin{x 1, x 2 } 2 λ 1 + λ 2 = 1 λ 1, λ R λ 2 = 1 λ 1 λ 1 x 1 + λ 2 x 2 = λ 1 x 1 + (1 λ 1 )x 2 = λ 1 (x 1 x 2 ) + x 2.

3 λ 1 R aff{x 1, x 2 } x 1 x 2 x 1, x 2 x 1, x 2 x 1, x 2 1.8 1.7 k 2 x 1,..., x k E n x 1,..., x k 1.8 E n k 2 x 1,..., x k E n x 1,..., x k x 1,..., x k x 1, x 2 x 1 x 2 x 1, x 2 2 x 1, x 2 2 x 1, x 2, x 3 x 1 x 2 x 3 x 1, x 2 2 x 1, x 3 2 x 1, x 2, x 3 2 x 1 x 2, x 3 x 1 x 2, x 3 x 1, x 2, x 3 2 x 1, x 2, x 3 3 x 1, x 2, x 3 3 x 1, x 2, x 3, x 4 x 1, x 2, x 3, x 4 4 2 3 4 1.9 k 2 x 1,..., x k E n (1) x 1,..., x k (2) λ i R λ 1 x 1 + +λ k x k = 0, λ 1 + +λ k = 0 λ 1,..., λ k = 0 (3) x 2 x 1,..., x k x 1 (4) τ : E n E n R τ(x) = (x, 1) τ(x 1 ),..., τ(x k )

4 1. (1) (2) x 1,..., x k x k x 1,..., x k 1 k 1 x k = µ i x i, k 1 µ i = 1 µ 1,..., µ k 1 1 0 k 1 µ i x i x k = 0, k 1 µ i 1 = 0 λ 1 = µ 1,..., λ k 1 = µ k 1, λ k = 1 (2) λ i x i = 0 λ i = 0 0 λ i R λ k 0 x k = k 1 λ i λ k x i, k 1 λ i λ k = 1 x k x 1,..., x k 1 x 1,..., x k (1) (2) (2) (4) τ(x 1 ),..., τ(x k ) λ 1 τ(x 1 ) + + λ k τ(x k ) = 0 (λ i R) τ (λ 1 x 1 + + λ k x k, λ 1 + + λ k ) = (0, 0) E n R (2) λ 1 = = λ k = 0. τ(x 1 ),..., τ(x k ) (4) (3) x 2 x 1,..., x k x 1 λ 2 (x 2 x 1 ) + + λ k (x k x 1 ) = 0 (λ i R) x 1,..., x k ( λ 2 λ k )x 1 + λ 2 x 2 + + λ k x k = 0

5 λ 1 = λ 2 λ k λ 1 τ(x 1 ) + + λ k τ(x k ) =(λ 1 x 1 + + λ k x k, λ 1 + + λ k ) = (0, 0). (4) λ 1 = = λ k = 0 x 2 x 1,..., x k x 1 (3) (2) (2) λ 1 x 1 + + λ k x k = 0, λ 1 + + λ k = 0 2 λ 1 = λ 2 λ k ( λ 2 λ k )x 1 + λ 2 x 2 + + λ k x k = 0 λ 2 (x 2 x 1 ) + + λ k (x k x 1 ) = 0 (3) λ 2 = = λ k = 0 λ 1 0 1.10 V E n x i V λ i R λ 1 x 1 + + λ k x k V 1.11 E n A x i V λ 1 + + λ k = 1 λ i R λ 1 x 1 + + λ k x k A 1.12 A E n x 0 A A x 0 = {x x 0 x A} E n A x 0 x 0 A dim A = dim(a x 0 )

6 1. u, v A x 0 x, y A u = x x 0, v = y x 0 u + v = x x 0 + y x 0 = (x + y x 0 ) x 0 x + y x 0 x, y, x 0 A A A u + v A x 0 λ R λu = λ(x x 0 ) = λx + (1 λ)x 0 x 0 λx + (1 λ)x 0 x, x 0 A A λu A x 0 A x 0 y 0 A u A x 0 x A u = x x 0 u = x x 0 + y 0 y 0 x x 0 + y 0 x, x 0, y 0 A A u A y 0 A x 0 A y 0 A y 0 A x 0 A x 0 = A y 0 A x 0 x 0 A 1.13 V E m W E n f : V W x i V λ i R f(λ 1 x 1 + + λ k x k ) = λ 1 f(x 1 ) + + λ k f(x k ) 1.14 A E m S E n f : A B x i A λ 1 + + λ k = 1 λ i R f(λ 1 x 1 + + λ k x k ) = λ 1 f(x 1 ) + + λ k f(x k )

7 1.15 A E m S E n f : A B x 0 A f l (u) = f(u + x 0 ) f(x 0 ) (u A x 0 ) f l : A x 0 B f(x 0 ) f l x 0 A u, v A x 0 x, y A u = x x 0, v = y x 0 λ R f l (u + v) =f(u + v + x 0 ) f(x 0 ) = f(x + y x 0 ) f(x 0 ) (x + y x 0 x, y, x 0 ) =f(x) + f(y) f(x 0 ) f(x 0 ) =f l (u) + f l (v). f l (λu) =f(λu + x 0 ) f(x 0 ) = f(λ(x x 0 ) + x 0 ) f(x 0 ) =f(λx + (1 λ)x 0 ) f(x 0 ) (λx + (1 λ)x 0 x, x 0 A ) =λf(x) + (1 λ)f(x 0 ) f(x 0 ) = λ(f(x) f(x 0 )) =λf l (u). f l : A x 0 B f(x 0 ) x 0 A u A x 0 x A u = x x 0 f(u + y 0 ) f(y 0 ) =f(x x 0 + y 0 ) f(y 0 ) (x x 0 + y 0 x, x 0, y 0 A ) =f(x) f(x 0 ) + f(y 0 ) f(y 0 ) = f(u + x 0 ) f(x 0 ). f l x 0 A 2 E n E n d d(x, y) = x y (x, y E n ). d

8 2. (1) x, y E n d(x, y) 0 d(x, y) = 0 x = y (2) x, y E n d(x, y) = d(y, x) (3) x, y, z E n d(x, z) d(x, y) + d(y, z). d(x, y) x, y E n d (1) (3) (1) (3) d 2.1 X d : X X R d X (X, d) (1) x, y X d(x, y) 0 d(x, y) = 0 x = y (2) x, y X d(x, y) = d(y, x) (3) x, y, z X d(x, z) d(x, y) + d(y, z). z X ρ > 0 B(z, ρ) = {x X d(z, x) ρ}, B 0 (z, ρ) = {x X d(z, x) < ρ} z ρ B(z, ρ) B 0 (z, ρ) d E n (E n, d) E n 2.2 (X, d) O X O X z O ρ > 0 B(z, ρ) O C X C 2.3 (X, d) O C O

9 (1) X O O (2) O 1, O 2 O O 1 O 2 O (3) O (O λ ) λ Λ λ Λ O λ O C (1) X C C (2) C 1, C 2 C C 1 C 2 C (3) C (C λ ) λ Λ C λ C λ Λ 2.4 (X, d) A A A inta A A cla inta A cla cla inta A bda A E n relinta, relbda (X, d) A 2.2 inta = O, cla = C. O A,O O A C,C C 2.5 (X, d) f f x 0 X ɛ > 0 δ > 0 x X, d(x, x 0 ) δ f(x) f(x 0 ) ɛ f X f R x y x, y (X, d) (X, d ) F F x 0 X ɛ > 0 δ > 0 x X, d(x, x 0 ) δ d (F (x), F (x 0 )) ɛ F X F

10 2. 2.6 (X, d) (X, d ) F X O F 1 (O ) X 2.2 2.2 2.7 (X, d) (X, d ) F F ɛ > 0 δ > 0 x, y X, d(x, y) δ d (F (x), F (y)) ɛ ɛ δ X L 0 F F L-Lipschitz x, y X d (F (x), F (y)) Ld(x, y). L 0 L-Lipschitz Lipschitz Lipschitz x x 2 [0, ) [0, ) x x 1/2 Lipschitz 2.8 (X, d) A A X (O λ ) λ Λ A λ Λ O λ Λ λ 1, λ 2,..., λ k Λ A k O λi 2.9 (Heine-Borel) E n A A A

11 3 x, y E n [x, y] = {(1 λ)x + λy 0 λ 1} [x, y) = {(1 λ)x + λy 0 λ < 1} A, B E n λ R A + B = {a + b a A, b B}, λa = {λa a A} A + B λa ( 1)A A A + ( B) A B x E n A + {x} A + x 3.1 A E n x, y A [x, y] A 3.2 f : E m E n x, y E m 0 λ 1 (1 λ)f(x) + λf(y) = f((1 λ)x + λy) [f(x), f(y)] = f([x, y]) 3.3 A, B A + B λ R λa A λ E n (λ Λ) x, y λ Λ A λ λ Λ x, y A λ [x, y] A λ [x, y] λ Λ A λ λ Λ A λ f : E m E n A E m u, v f(a) E n x, y A u = f(x), v = f(y) 3.2 [u, v] = [f(x), f(y)] = f([x, y]) f(a) f(a) B E n x, y f 1 (B) f(x), f(y) B 3.2 f([x, y]) = [f(x), f(y)] B [x, y] f 1 (B) f 1 (B)

12 3. A B E n E n = E 2n (a 1, b 1 ), (a 2, b 2 ) 0 λ 1 (1 λ)(a 1, b 1 ) + λ(a 2, b 2 ) = ((1 λ)a 1 + λa 2, (1 λ)b 1 + λb 2 ) [a 1, a 2 ] [b 1, b 2 ] [(a 1, b 1 ), (a 2, b 2 )] [a 1, a 2 ] [b 1, b 2 ] A B A B f : E n E n E n ; (x, y) x + y f f f A + B = f(a B) A + B 3.4 A E n λ, µ > 0 (λ + µ)a λa + µa A (λ + µ)a = λa + µa x (λ + µ)a x = (λ + µ)a a A x = (λ + µ)a = λa + µa λa + µa (λ + µ)a λa + µa A x λa + µa a, b A x = λa + µb λ, µ > 0 ( λ x = (λ + µ) λ + µ a + µ ) λ + µ b (λ + µ)a (λ + µ)a = λa + µa 3.5 x 1,..., x k E n λ 1,..., λ k R x = λ i x i (λ i 0 (i = 1,..., k)), λ i = 1 x 1,..., x k A E n A conva A 3.6 A E n conva = A A E n conva = {K K, A K} A, B E n conv(a + B) = conva + convb

13 A A conva A k A k k = 1 A A k 1 A A k A A x 1,..., x k A λ 1 + +λ k = 1 λ 1,..., λ k 0 x = λ 1 x 1 + + λ k x k λ k = 1 x = x k A λ k < 1 k 1 λ i x = (1 λ k ) x i + λ k x k 1 λ k k 1 λ i 1 λ k = 1, k 1 λ i 1 λ k 0 λ i 1 λ k x i A. x x A A A conva A conva = A A E n conva x, y conva A ) l l x = λ i a i, y = µ j b j (a i, b j A, λ i, µ j 0, λ i = µ j = 1 j=1 0 λ 1 j=1 l (1 λ)x + λy = (1 λ) λ i a i + λ µ j b j = j=1 l (1 λ)λ i a i + λµ j b j j=1 (1 λ)λ i 0, λµ j 0, l (1 λ)λ i + λµ j = (1 λ) + λ = 1 j=1 (1 λ)x + λy A (1 λ)x + λy conva conva D(A) = {K K, A K}

14 3. A conva conva D(A) conva A K K conva convk = K conva D(A) conva = D(A) = {K K, A K} A, B E n x conv(a + B) ( x = λ i (a i + b i ) a i A, b i B, λ i 0, ) λ i = 1 x = λ i a i + λ i b i conva + convb conv(a + B) conva + convb x conva + convb ) l l x = λ i a i + µ j b j (a i A, b j B, λ i, µ j 0, λ i = µ j = 1 j=1 j=1 l λ i µ j (a i + b j ) = j=1 = l l λ i µ j a i + λ i µ j b j j=1 λ i a i + j=1 j=1 l µ j b j = x l λ i µ j = j=1 λ i l µ j = 1 λ i µ j 0 x A+B a i +b j x conv(a+b) conva + convb conv(a + B) j=1 conv(a + B) = conva + convb 3.7 k + 1 k k + 1

15 3.8 A E n x inta y cla [x, y) inta 0 < λ < 1 z = (1 λ)y + λx x inta ρ > 0 B(x, ρ) A y A B(z, λρ) A w B(z, λρ) w = (w z) + z = (w z) + (1 λ)y + λx = (1 λ)y + λ (x + 1λ ) (w z) 1 (w z) λ = 1 w z ρ λ x + 1 (w z) B(z, λρ) A. λ A w A B(z, λρ) A z inta [x, y) inta y cla V = B(y, λρ/(1 λ)) y cla A V a A V z = (1 λ)y + λx = (1 λ)(y + (a y)) + λx (1 λ)(a y) ( = (1 λ)a + λ x 1 λ ) (a y) λ 1 λ (a y) λ = 1 λ a y ρ λ x 1 λ (a y) B(x, ρ) A. λ A z A [x, y) A [x, y) inta y 1 [x, y) y 1 y y 2 [x, y) [x, y 1 ) [x, y 2 ) [x, y) A y 2 A [x, y 2 ) inta y 1 inta [x, y) inta 3.9 λ 1,..., λ k B(x 1, ρ 1 ),..., B(x k, ρ k ) ( ) λ i B(x i, ρ i ) = B λ i x i, λ i ρ i.

16 3. x E n ρ 0 B(x, ρ) = B(0, ρ) + x. λ R λb(x, ρ) = λ(b(0, ρ) + x) = B(0, λ ρ) + λx. ( ) λ i B(x i, ρ i ) = (B(0, λ i ρ i ) + λ i x i ) = B 0, λ i ρ i + ( = B λ i x i, ) λ i ρ i. λ i x i 3.10 A E n cla = ɛ>0(a + B(0, ɛ)) x cla ɛ > 0 B(x, ɛ) A y B(x, ɛ) A x B(y, ɛ) y A x B(y, ɛ) = y + B(0, ɛ) A + B(0, ɛ). ɛ > 0 x A + B(0, ɛ) cla ɛ>0(a + B(0, ɛ)) x ɛ>0(a + B(0, ɛ)) ɛ > 0 x A + B(0, ɛ) y A x y + B(0, ɛ) = B(y, ɛ) y B(x, ɛ) A B(x, ɛ) A x cla (A + B(0, ɛ)) cla ɛ>0 3.11 A E n inta cla A conva

17 3.8 x, y inta [x, y] inta inta 3.10 cla = ɛ>0(a + B(0, ɛ)) 3.3 3.6 A x conva ) x = λ i x i (λ i > 0, λ i = 1, x i A A x i ρ i > 0 B(x i, ρ i ) A 3.9 ( ) B λ i x i, λ i ρ i = λ i B(x i, ρ i ) conva conva 3.12 A E n (1) relinta = relint cla, (2) cla = cl relinta, (3) relbda = relbd cla = relbd relinta. affa A E n inta relinta = inta, relbda = bda (1) A cla inta int cla x int cla x y inta y x x int cla z z cla 3.8 [y, z) inta x [y, z) x inta int cla inta inta = int cla (2) inta A cl inta cla x cla x y inta 3.8 [y, x) inta x cl inta cla cl inta cla = cl inta (3) (1) (2) bd cla = cl cla \ int cla = cla \ inta = bda. bd inta = cl inta \ int inta = cla \ inta = bda. 3.13 E n E n K n

18 4. 4 A E n 4.1 x E n x z x y (y A) z A ρ > 0 B(x, ρ) A B(x, ρ) A y x y y 0 B(x, ρ) A y A x y 0 x y y 1 A y A x y 1 x y y 0 y 1 A y x y x y 0 = x y 1 z = (y 0 + y 1 )/2 A z A y 0 y 1 z xy 0 y 1 x z < x y 0 y 0 A 4.2 4.1 z x A z = p(a, x) x E n \ A x p(a, x) = d(a, x) u(a, x) = x p(a, x) d(a, x) p(a, x) x R(A, x) = {p(a, x) + λu(a, x) λ 0} p(a, x) u(a, x) 4.3 x E n \ A y R(A, x) p(a, x) = p(a, y) p(a, y) p(a, x) y = p(a, x) y A p(a, y) = y = p(a, x) y R(A, x) \ {p(a, x)} y [x, p(a, x)) p(a, y) p(a, x) A 4.1 y p(a, y) < y p(a, x) x p(a, y) x y + y p(a, y) < x y + y p(a, x) = x p(a, x) p(a, x) x [y, p(a, x)) [p(a, x), p(a, y)] q [x, q] [y, p(a, y)] x q x p(a, x) = y p(a, y) y p(a, x) < 1 x q < x p(a, x) p(a, x) p(a, y) = p(a, x)

19 4.4 1-Lipschitz p(a, x) p(a, y) x y (x, y E n ). v = p(a, x) p(a, y) v 0 ( ) x p(a, x), v 0 x p(a, x), v < 0 x A R(A, x) x p(a, x), v < 0 p(a, y) v R(A, x) z z p(a, y) v z p(a, y) = z p(a, x) 2 v 2 < z p(a, x). z R(A, x) 4.3 p(a, x) = p(a, z) z p(a, y) < z p(a, z) p(a, z) ( ) ( ) x y y p(a, y), p(a, y) p(a, x) 0 y p(a, y), v 0 ( ) p(a, x), p(a, y) [x, y] p(a, x), p(a, y) p(a, x) p(a, y) x y 4.5 S A p(a, S) = bda S A p(a, S) bda bda p(a, S) x bda i N x x i < 1/i A S x i 4.4 x p(a, x i ) = p(a, x) p(a, x i ) x x i < 1/i. R(A, x i ) S y i 4.3 p(a, y i ) = p(a, x i ) x p(a, y i ) < 1/i (y i ) i N S y S (y ij ) j N 4.4 p(a, ) x = lim j p(a, y ij ) = p(a, lim j y ij ) = p(a, y) p(a, S). bda p(a, S) p(a, S) = bda

20 5. 5 E n u E n \ {0} α R H u,α = {x E n x, u = α} H u,α H u,α = {x E n x, u α}, H + u,α = {x E n x, u α}. 5.1 H E n H +, H A E n x A x A H A H + A H H x A H A x A H A H A H = H u,α A A Hu,α H u,α A u H u,α Hu,α H x A u A x 5.2 A E n x E n \ A p(a, x) u(a, x) A p(a, x) u(a, x) H p(a, x) H A H x H A H y A y H [p(a, x), y] x z y H y, u(a, x) > p(a, x), u(a, x) y p(a, x), u(a, x) > 0 z p(a, x) x z < x p(a, x) [p(a, x), y] A z A p(a, x) y A A H H A 5.3 A E n A A A u E n \ {0} u A x bda A 4.5 x = p(a, y) y E n \ A 5.2 x y x x A A A B(x, 1) x x A B(x, 1) H H x A B(x, 1) A H z A \ H z, x A

21 [z, x] A z H x H = bdh [z, x) H [z, x) B(x, 1) A B(x, 1) H A H H x A A u E n \ {0} A x A x, u = sup{ y, u y A} {y E n y, u = x, u } u A 5.4 A, B E n H u,α E n A Hu,α B H u,α + A H+ u,α B H u,α H A B A inthu,α B inth+ u,α A inth u,α + B inth u,α H A B ɛ > 0 H u,α ɛ H u,α+ɛ A B H A B A x A {x} 5.5 A E n x E n \ A A x A A x A 5.2 p(a, x) u(a, x) A A x [p(a, x), x] (p(a, x) + x)/2 A x A x cla cla x A x x cla x A x bda 3.12 bda = bd cla x bd cla 3.11 cla 5.3 x cla cla x A x 5.6 E n A E n A S(A) A A A S(A) A x 5.5 A x A x S(A) A = S(A)

22 6. 6 R = R {, } ± λ R + = λ + = + λ =, = λ + ( ) = λ = + λ =, (λ > 0), λ = λ = 0 (λ = 0), (λ < 0). 6.1 f : E n R f 1 ( ) = f 1 ( ) E n f((1 λ)x + λy) (1 λ)f(x) + λf(y) (x, y E n, 0 λ 1) f D E n f : D R { f(x) (x D) f(x) = (x E n \ D) f f f f 6.2 III III III III III 6.3 u E n α R f(x) = u, x + α f((1 λ)x + λy) = u, (1 λ)x + λy + α = (1 λ)( u, x + α) + λ( u, y + α) = (1 λ)f(x) + λf(y) 6.4 (Jensen ) f : E n R x 1,..., x k E n λ 1 + + λ k = 1 λ 1,..., λ k [0, 1] f(λ 1 x 1 + + λ k x k ) λ 1 f(x 1 ) + + λ k f(x k ) Jensen

23 k k = 1 λ 1 = 1 k = 2 k 1 k λ k = 1 λ k < 1 ( ) k 1 λ i f(λ 1 x 1 + + λ k x k ) = f (1 λ k ) x i + λ k x k 1 λ k ( k 1 ) λ i (1 λ k )f x i + λ k f(x k ) 1 λ k 6.5 f : E n R k 1 λ i (1 λ k ) f(x i ) + λ k f(x k ) 1 λ k = λ 1 f(x 1 ) + + λ k f(x k ). domf = f 1 ((, )) f domf epif = {(x, ζ) E n R f(x) ζ} f epif A E n { 0 (x A) I A (x) = (x E n \ A) A I A 6.6 f : E n R f domf α R f 1 ((, α)) f 1 ((, α]) epif A E n I A x, y domf 0 λ 1 f((1 λ)x + λy) (1 λ)f(x) + λf(y) < (1 λ)x + λy domf domf x, y f 1 ((, α)) 0 λ 1 f((1 λ)x + λy) (1 λ)f(x) + λf(y) < (1 λ)α + λα = α

24 6. (1 λ)x + λy f 1 ((, α)) f 1 ((, α)) f 1 ((, α]) (x, ζ), (y, η) epif 0 λ 1 (1 λ)(x, ζ) + λ(y, η) = ((1 λ)x + λy, (1 λ)ζ + λη) f((1 λ)x + λy) (1 λ)f(x) + λf(y) (1 λ)ζ + λη (1 λ)(x, ζ) + λ(y, η) epif epif A E n I A 0 λ 1 x, y A (1 λ)x + λy A I A ((1 λ)x + λy) = 0 = (1 λ)i A (x) + λi A (y). x A y E n \ A 0 < λ λ = 0 x, y E n \ A (1 λ)i A (x) + λi A (y) = I A ((1 λ)x + λy). (1 λ)i A (x) + λi A (y) = I A (x) = I A ((1 λ)x + λy). (1 λ)i A (x) + λi A (y) = I A ((1 λ)x + λy). I A 6.7 f : E n R int domf int domf f Lipschitz x 0 int domf f x 0 ints S int domf S x 0 ints B(x 0, ρ) S ρ > 0 S x 1,..., x n+1 x S ) n+1 n+1 x = λ i x i (λ i 0, λ i = 1 c = max{f(x 1 ),..., f(x n+1 )} Jensen ( 6.4) n+1 f(x) λ i f(x i ) c.

25 B(x 0, ρ) y α [0, 1] u = ρ u y = x 0 +αu y x 0 x 0 + u y = (1 α)x 0 + α(x 0 + u) f(y) (1 α)f(x 0 ) + αf(x 0 + u) x 0 + u S 0 α f(y) f(x 0 ) α(f(x 0 + u) f(x 0 )) α(c f(x 0 )). x 0 y x 0 u x 0 = (1 λ)y + λ(x 0 u) x 0 = (1 λ)(x 0 + αu) + λ(x 0 u) = (1 λ)x 0 + (1 λ)αu + λx 0 λu = x 0 + (α λα λ)u = x 0 + (α λ(1 + α))u λ = α/(1 + α) 1 + α x 0 = 1 1 + α y + α 1 + α (x 0 u) f(x 0 ) 1 1 + α f(y) + α 1 + α f(x 0 u) (1 + α)f(x 0 ) f(y) + αf(x 0 u). f(x 0 ) f(y) α(f(x 0 u) f(x 0 )) α(c f(x 0 )). f(y) f(x 0 ) α(c f(x 0 )). y x 0 = αu = αρ α = y x 0 /ρ f(y) f(x 0 ) 1 ρ (c f(x 0)) y x 0 (y B(x 0, ρ)) f x 0 f int domf C int domf f Lipschitz C ρ > 0 C ρ = C + B(0, ρ)

26 7. int domf C ρ f C ρ a x, y C (x y) z = y + ρ (y x) y x z C ρ y x z y = (1 λ)x + λz ( y = (1 λ)x + λ y + ρ ) (y x) y x ( y x + ρ = (1 λ)x + λ y ρ ) y x y x x (1 λ) y x λ y x + ρ = x + λ y y x y x y x λ( y x + ρ) y x + ρ = x + λ y. y x y x λ = y = (1 λ)x + λz y x y x + ρ f(y) (1 λ)f(x) + λf(z) y x f(y) f(x) λ(f(z) f(x)) = (f(z) f(x)) y x + ρ y x y x + ρ 2a 2a y x ρ x = y x, y C f(y) f(x) 2a ρ y x C f Lipschitz 7 7.1 K E n K h(k, ) = h K h(k, u) = sup{ x, u x K} (u E n )

27 u domh(k, ) \ {0} H(K, u) = {x E n x, u = h(k, u)}, H (K, u) = {x E n x, u h(k, u)} H(K, u) K H (K, u) K 7.2 K E n h(k + t, u) = h(k, u) + t, u (t, u E n ), h(k, λu) = h(λk, u) = λh(k, u) (λ 0, u E n ), h(k, u + v) h(k, u) + h(k, v) (u, v E n ) K E n h(k, ) int domh K L E n h K h L K L h(k + t, u) = sup{ x + t, u x K} = sup{ x, u + t, u x K} = h(k, u) + t, u, h(k, λu) = sup{ x, λu x K} = sup{λ x, u x K} = λh(k, u), h(λk, u) = sup{ λx, u x K} = sup{λ x, u x K} = λh(k, u), h(k, u + v) = sup{ x, u + v x K} 0 λ 1 = sup{ x, u + x, v x K} sup{ x, u x K} + sup{ x, v x K} = h(k, u) + h(k, v). h(k, (1 λ)u + λv) h(k, (1 λ)u) + h(k, λv) = (1 λ)h(k, u) + λh(k, v) h(k, ) 6.7 h(k, ) int domh K

28 7. K L h K (u) = sup{ x, u x K} sup{ x, u x L} = h L (u) u E n h K (u) h L (u) K L 5.6 K L 7.3 K, L K n 3.3 K + L K n Minkowski K n f f(k + L) = f(k) + f(l) (K, L K n ) f Minkowski 7.4 K, L K n h(k + L, ) = h(k, ) + h(l, ). 0 0 E n \ {0} K, L u E n \{0} x K h(k, u) = x, u y L h(l, u) = y, u h(k, u) + h(l, u) = x + y, u h(k + L, u) z K + L z = x + y x K y L z, u = x, u + y, u h(k, u) + h(l, u) h(k + L, u) h(k, u) + h(l, u) h(k + L, u) = h(k, u) + h(l, u) (u E n ). 7.5 (K n, +) {0} K, L, M K n λ, µ 0 (1) K + M = L + M K = L, (2) λ(k + L) = λk + λl, (3) (λ + µ)k = λk + µk, (4) λ(µk) = (λµ)k, (5) 1K = K.

29 (1) K + M = L + M h(k, ) + h(m, ) = h(k + M, ) = h(l + M, ) = h(l, ) + h(m, ) h(k, ) = h(l, ) K L 5.6 K = L (2) (4) (5) (3) 7.2 (1) h((λ + µ)k, ) = (λ + µ)h(k, ) = λh(k, ) + µh(k, ) = h(λk, ) + h(µk, ) = h(λk + µk, ) (λ + µ)k = λk + µk 8 Hausdorff 8.1 E n C n K, L C n δ(k, L) = max{sup{d(x, L) x K}, sup{d(x, K) x L}} Hausdorff 8.2 K, L C n δ(k, L) = min{λ 0 K L + λb n, L K + λb n } δ C n α = min{λ 0 K L + λb n, L K + λb n } x K x L + αb n y L b B n x = y + αb x y = αb α d(x, L) α x K sup{d(x, L) x K} α. L K sup{d(x, K) x L} α. δ(k, L) α 0 < λ < α λ K L + λb n L K + λb n K L + λb n x K x L + λb n y L x y > λ d(x, L) λ δ(k, L) λ L K + λb n δ(k, L) λ 0 < λ < α λ δ(k, L) λ δ(k, L) α δ(k, L) = α

30 8. Hausdorff δ K, L C n δ(k, L) = δ(l, K) 0 x K d(x, K) = 0 δ(k, K) = 0 δ(k, L) = 0 x K d(x, L) = 0 x L x L d(x, K) = 0 x K K = L K, L, M C n δ(k, L) = α, δ(l, M) = β K L + αb n L M + βb n K M + αb n + βb n = M + (α + β)b n δ(k, M) α + β = δ(k, L) + δ(l, M). 8.3 (K i ) i N C n i N K i+1 K i C n Hausdorff (K i ) i N K = K i K lim K i K ɛ > 0 m N i δ(k m, K) > ɛ K m K + ɛb n A m = K m \int(k +ɛb n ) A m A = A m m N m=1 A m int(k + ɛb n ) = A int(k + ɛb n ) = A K = A m K m A K A lim i K i = K 8.4 (C n, δ) (K i ) i N C n Cauchy (K i ) i N R > 0 δ(k 1, K i ) R K i K 1 +RB n ( ) i N K i A m = cl K i C n 8.3 lim m A m = A := m=1 i=m A m ɛ > 0 n 0 N m n 0 δ(a, A m ) ɛ

31 A m A + ɛb n i n 0 K i A + ɛb n (K i ) i N Cauchy n 1 n 0 i, j n 1 K j K i + ɛb n i, m n 1 K j K i + ɛb n K i + ɛb n A m K i + ɛb n A i n 1 A K i + ɛb n i n 1 δ(k i, A) ɛ K i A (C n, δ) 8.5 C n (Ki 0 ) i N C n i Ki 0 En C γ m N C 2 m C 2 m γ 2 mn K C n K A m (K) (Ki 0 ) i N (Ki 1 ) i N A 1 (Ki 1 ) = T 1 i N (Ki 1 ) i N (Ki 2 ) i N A 2 (Ki 2 ) = T 2 i N m N (Ki m ) i N ( ) A m (Ki m ) = T m (i N) ( ) j=m k < m (K m i ) i N (K k i ) i N ( ) i, j N A m (Ki m ) = A m (Kj m ) = T m T m Ki m Kj m K i m Kj m + 2 m nγb n i, j, m N δ(ki m, Kj m ) 2 m nγ ( ) i, j N k m δ(ki m, Kj k ) 2 m nγ K m = Km m (K m) m N (Ki 0 ) i N δ(k m, K k ) 2 m nγ (k m). (K m ) m N Cauchy 8.4 8.6 K n C n (K n, δ) C n \K n K C n \K n K x, y K 0 < λ < 1 z = (1 λ)x + λy K K ɛ > 0 B(z, ɛ) K = C n K ɛ/2 K K C n δ(k, K ) < ɛ/2 x, y K d(x, K ) < ɛ/2, d(y, K ) < ɛ/2 x x < ɛ/2, y y < ɛ/2 x, y K z = (1 λ)x + λy z z = (1 λ)(x x) + λ(y y) (1 λ) x x + λ y y < ɛ/2.

32 8. Hausdorff z K δ(k, K ) < ɛ/2 w z < ɛ/2 w K w z w z + z w < ɛ B(z, ɛ) K = z K K K ɛ/2 C n \ K n C n \ K n K n C n 8.4 C n K n 8.7 E n 8.5 8.6 K n C n K n 8.8 K i, K K n lim i K i = K (1) (2) (1) K i N x i K i (x i ) i N (2) j N x ij K ij (i j ) j N (x ij ) j N (x ij ) j N K lim K i = K lim δ(k, K i ) = 0 i i K x x i = p(k i, x) x i K i x x i = d(x, K i ) δ(k, K i ) x i i x (1) j N x ij K ij (i j ) j N (x ij ) j N x = lim x ij K K j ρ > 0 B(x, ρ) (K + ρb n ) = x = lim x ij j j x ij x < ρ lim δ(k, K i ) = 0 i j δ(k ij, K) < ρ x ij K ij K + ρb n x ij B(x, ρ) (K + ρb n ) x K (2) (1) (2) ɛ > 0 ( ) ( ) i K K i + ɛb n i K i K + ɛb n lim i K i = K ( ) (k j ) K K kj + ɛb n y kj K y kj (K kj + ɛb n ) y kj y kj d(y kj, K kj ) ɛ K (y kj ) y ij y ij i y y K y (1) x i K i x i i y y ij x ij

33 j y x ij y ij 0 x ij K ij d(y kj, K kj ) ɛ ( ) ( ) (i j ) K ij K + ɛb n y ij K ij y ij K + ɛb n y ij (1) K K i j x ij K ij x ij K + ɛb n x ij x ij, y ij K ij, x ij K + ɛb n, y ij K + ɛb n x ij y ij z ij K ij bd(k + ɛb n ) z ij bd(k + ɛb n ) z ij bd(k + ɛb n ) (2) K ( ) 8.9 K, L C n δ(convk, convl) δ(k, L) conv : C n K n ; K convk Lipschitz K, K, L, L C n δ(k + K, L + L ) δ(k, L) + δ(k, L ), δ(k K, L L ) max{δ(k, L), δ(k, L )} + : C n C n C n ; (K, K ) K + K, : C n C n C n ; (K, K ) K K C n C n Lipschitz K L + λb n K L + λb n convl + λb n. 3.6 convl + λb n convk convl + λb n L K + λb n convl convk + λb n δ(convk, convl) λ δ(convk, convl) δ(k, L) K L + λb n K L + µb n K + K L + L + λb n + µb n = (L + L ) + (λ + µ)b n, K K (L + λb n ) (L + µb n ) (L L ) + max{λ, µ}b n

34 8. Hausdorff L K + λb n L K + µb n L + L (K + K ) + (λ + µ)b n, L L (K K ) + max{λ, µ}b n δ(k + K, L + L ) λ + µ, δ(k K, L L ) max{λ, µ} δ(k + K, L + L ) δ(k, L) + δ(k, L ), δ(k K, L L ) max{δ(k, L), δ(k, L )} 8.10 K, L K n δ(k, L) = sup{ h(k, u) h(l, u) u S n 1 }. δ(k, L) α Hausdorff K L + αb n 7.2 u S n 1 h(k, u) h(l + αb n, u) = h(l, u) + α h(k, u) h(l, u) α δ(k, L) α L K + αb n h(k, u) h(l, u) α δ(k, L) α sup{ h(k, u) h(l, u) u S n 1 } α sup{ h(k, u) h(l, u) u S n 1 } δ(k, L) sup{ h(k, u) h(l, u) u S n 1 } α u S n 1 h(k, u) h(l, u) α h(k, u) h(l, u) α h(k, u) h(l, u) + α = h(l + αb n, u). 7.2 K L + αb n h(l, u) h(k, u) α L K + αb n δ(k, L) α δ(k, L) sup{ h(k, u) h(l, u) u S n 1 } δ(k, L) = sup{ h(k, u) h(l, u) u S n 1 }

35 8.11 K 1, K 2 K n K 2 intk 1 η > 0 δ(k 1, K) < η K K n K 2 K K 2 u E n \ {0} x 2 K 2 h(k 2, u) = x 2, u K 2 intk 1 ρ > 0 x 1 = x 2 + ρu/ u 2 intk 1 K 1 x 1, u = x 2 + ρ u u, u = x 2 2, u + ρ = h(k 2, u) + ρ h(k 1, u) x 1, u = h(k 2, u) + ρ > h(k 2, u). E n \{0} h(k 1, ) h(k 2, ) > 0 h(k 1, ), h(k 2, ) 7.2 6.7 h(k 1, ) h(k 2, ) S n 1 η η h(k 1, u) h(k 2, u) (u S n 1 ) δ(k 1, K) < η K K n 8.10 h(k 1, u) h(k, u) < η (u S n 1 ) u S n 1 h(k 2, u) h(k 1, u) η < h(k, u) 7.2 K 2 K 8.12 V n K n K 0 K n V n (K 0 ) = 0 K 0 E n H δ(k 0, K) = α 1 K K 0 + αb n H u I = [ u, u] K 0 + αb n H (K 0 + αb n ) + αi H (K 0 + B n ) + αi Fubini V n (H (K 0 + B n ) + αi) = V n 1 (H (K 0 + B n ))2α. V n (K) 2V n 1 (H (K 0 + B n ))α = 2V n 1 (H (K 0 + B n ))δ(k 0, K)

36 8. Hausdorff V n K 0 V n (K 0 ) > 0 K 0 0 intk 0 ρb n intk 0 ρ > 0 ɛ > 0 (λ n 1)λ n V n (K 0 ) < ɛ λ > 1 ρb n intk 0 8.11 η > 0 δ(k 0, K) < η K K n ρb n K α = min{η, (λ 1)ρ} δ(k 0, K) < α K K n K 0 K + αb n K + (λ 1)ρB n K + (λ 1)K = λk. 7.5 (3) K λk 0 V n (K 0 ) V n (λk) = λ n V n (K), V n (K) λ n V n (K 0 ) V n (K 0 ) V n (K) (λ n 1)V n (K) (λ n 1)λ n V n (K 0 ) < ɛ. λ > 1 V n (K) V n (K 0 ) (λ n 1)V n (K 0 ) (λ n 1)λ n V n (K 0 ) < ɛ. V n (K 0 ) V n (K) < ɛ V n K 0 [1] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, 1993 1 2 [1] Conventions and notation 3 8 [1] 1.1 Convex sets and combinations 1.2 The metric projection 1.3 Support and separation 1.5 Convex functions 1.7 The support function 1.8 The Hausdorff metric 1 Hausdorff