1. duality in mathematics Some concepts and properties Half-spaces: &=xE R": px 2c) Normal vecto:p∈界 Hyperplane:H={x∈界":px=c} K is convex closure VX eK and x∈K,p∈界"andc∈界,3px<C≤pX K is concave, k is the closed convex hull K=∩(=K) lecture 4 for Chu Kechen Honors Collegelecture4 for Chu Kechen Honors College 1.Duality in mathematics ▪ Some concepts and properties: • Half-spaces: • Normal vector: • Hyperplane: • K is convex closure: • K is concave, K* is the closed convex hull: { : } n H =   x px c n p { : } n H c =  = x px          K x K c c and , and , n x p px px * ( ) p K K =  H