Matching Tag: random-polytopes
Let d d d n n n β 1 , … , β n ≥ − 1 \beta_1,\ldots,\beta_n\geq -1 β 1 , … , β n ≥ − 1 P n , d β 1 , … , β n \mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n} P n , d β 1 , … , β n
Let N , d f i n N N,dfin\mathbb{N} N , df in N N ≥ d + 2 N\geq d+2 N ≥ d + 2 [ N ] = A ∪ B [N]=A\cup B [ N ] = A ∪ B P d ( A , B ) \mathbb{P}_d(A,B) P d ( A , B ) ( A , B ) (A,B) ( A , B ) N N N
For n ∈ N n\in\mathbb{N} n ∈ N N n N_n N n n n n P N n , d \bbbeta \mathcal{P}_{N_n,d}^{\bbbeta} P N n , d \bbbeta P n , d \bbbeta \mathcal{P}_{n,d}^{\bbbeta} P n , d \bbbeta
Let \bbbeta = ( β 1 , … , β m ) \bbbeta=(\beta_1,\ldots,\beta_m) \bbbeta = ( β 1 , … , β m ) P n , d \bbbeta \mathcal{P}_{n,d}^{\bbbeta} P n , d \bbbeta
Let d = d 1 + ⋯ + d m d=d_1+\cdots+d_m d = d 1 + ⋯ + d m \bbbeta \bbbeta \bbbeta P d , d \bbbeta \mathcal{P}_{d,d}^{\bbbeta} P d , d \bbbeta
Let m ∈ N m\in\mathbb{N} m ∈ N i ∈ 1 , … , m i\in\\{1,\dotsc,m\\} i ∈ 1 , … , m K i ⊂ R d i K_i\subset \mathbb{R}^{d_i} K i ⊂ R d i
Let P P P d d d 2 d 2d 2 d
Let Z d typ \mathcal Z_d^{\text{typ}} Z d typ f 0 ( Z d typ ) f_0(\mathcal Z_d^{\text{typ}}) f 0 ( Z d typ )
Fix k ∈ { 2 , 3 , … } k\in\{2,3,\ldots\} k ∈ { 2 , 3 , … } k k k conv Π d , 1 \operatorname{conv}\Pi_{d,1} conv Π d , 1
Let U 1 , … , U N U_1,\ldots,U_N U 1 , … , U N R d \mathbb{R}^d R d β > − 1 \beta>-1 β > − 1 B d = x ∈ R d : ∣ x ∣ ≤ 1 \mathbb B^d=\\{x\in\mathbb{R}^d:\\|x\\|\leq 1\\} B d = x ∈ R d : ∣ x ∣ ≤ 1 v m v_m v m
Let d ∈ N d\in\mathbb{N} d ∈ N k ∈ { 0 , … , d − 1 } k\in\{0,\ldots,d-1\} k ∈ { 0 , … , d − 1 } V d \mathcal{V}_d V d d d d E f k ( V d ) \mathbb{E}f_k(\mathcal{V}_d) E f k ( V d )
Let n ∈ N n\in\mathbb{N} n ∈ N k ∈ { 1 , … , n } k\in\{1,\ldots,n\} k ∈ { 1 , … , n } β > ( n − 1 ) / 2 \beta>(n-1)/2 β > ( n − 1 ) /2 J ~ n , k ( β ) \tilde{\mathbb{J}}_{n,k}(\beta) J ~ n , k ( β )
Let β ≥ − 1 \beta\geq -1 β ≥ − 1 n ∈ N n\in\mathbb{N} n ∈ N k ∈ { 1 , … , n } k\in\{1,\ldots,n\} k ∈ { 1 , … , n } J n , k ( β ) \mathbb{J}_{n,k}(\beta) J n , k ( β )
