Are the following sets convex? Justify your answer in each case. (a) (b) (c) (d) S...

Transcribed Image Text:

Are the following sets convex? Justify your answer in each case. (a) (b) (c) (d) S = {x R : a ax < }, where a = R and B = R are scalars and a R" is a normal vector. S = {x Rn : ai x Bi, i = 1,..., n}, where a; ER and i = R are scalars for all i = 1,..., n. S = = {x = R : ax b, a x b}, where a R and 2 R^ are normal vectors, and b ER and b ER are scalars. S = {x = R : x = Ay, y = Rm, y 0}, where A = Rmxn. Are the following sets convex? Justify your answer in each case. (a) (b) (c) (d) S = {x R : a ax < }, where a = R and B = R are scalars and a R" is a normal vector. S = {x Rn : ai x Bi, i = 1,..., n}, where a; ER and i = R are scalars for all i = 1,..., n. S = = {x = R : ax b, a x b}, where a R and 2 R^ are normal vectors, and b ER and b ER are scalars. S = {x = R : x = Ay, y = Rm, y 0}, where A = Rmxn.