Matching Tag: gaussian-geometry
Let n ≥ 2 n\geq2 n ≥ 2 ρ ∈ [ − 1 , 1 ] \rho\in[-1,1] ρ ∈ [ − 1 , 1 ] 3 ≤ k ≤ n + 1 3\leq k\leq n+1 3 ≤ k ≤ n + 1 { A i } i = 1 k \{A_i\}_{i=1}^k { A i } i = 1 k R n \mathbb{R}^n R n ⋃ i = 1 k A i = R n \bigcup_{i=1}^k A_i=\mathbb{R}^n ⋃ i = 1 k A i = R n
Let m ≥ 3 m\geq3 m ≥ 3 0 < ρ < 1 0<\rho<1 0 < ρ < 1 ( Ω i ) i = 1 m (\Omega_i)_{i=1}^m ( Ω i ) i = 1 m ( Ω i ′ ) i = 1 m (\Omega_i')_{i=1}^m ( Ω i ′ ) i = 1 m
Let m > 3 m>3 m > 3 a 1 , … , a m > 0 a_1,\dots,a_m>0 a 1 , … , a m > 0 ∑ i = 1 m a i = 1 \sum_{i=1}^m a_i=1 ∑ i = 1 m a i = 1 w ∈ R n + 1 w\in\mathbb{R}^{n+1} w ∈ R n + 1 w ‾ ( i ) = w / a i \overline{w}^{(i)}=w/a_i w ( i ) = w / a i
Barthe's conjecture. If Ω \Omega Ω r > 0 r>0 r > 0 0 ≤ k ≤ n 0\leq k\leq n 0 ≤ k ≤ n
Quadratic Symmetric Gaussian Problem. Either B ( 0 , r a ) B(0,r_a) B ( 0 , r a ) B ( 0 , r a ′ ) c B(0,r_a')^c B ( 0 , r a ′ ) c
Symmetric Gaussian Problem. If ρ > 0 \rho>0 ρ > 0 ( B ( 0 , r a ) , B ( 0 , r b ) c ) (B(0,r_a),B(0,r_b)^c) ( B ( 0 , r a ) , B ( 0 , r b ) c ) ( B ( 0 , r a ′ ) c , B ( 0 , r b ′ ) ) (B(0,r_a')^c,B(0,r_b')) ( B ( 0 , r a ′ ) c , B ( 0 , r b ′ ))
Equip R n \mathbb{R}^n R n
