√(x^2+y^2) ≥ (√2/2) * (x+y)
By 0: 00
√[x^2+( 1-y)^2]≥(√2/2)(x+ 1-y)
√[( 1-x)^2+y^2]≥(√2/2)( 1-x+y)
√[( 1-x)^2+( 1-y)^2]≥(√2/2)( 1-x+ 1-y)
Add up the four items:
√(x^2+y^2)+√[x^2+( 1-y)^2]+√[( 1-x)^2+y^2]+√[( 1-x)^2+( 1-y)^2]
≥(√2/2)[(x+y)+(x+ 1-y)+( 1-x+y)+( 1-x+ 1-y)]
= (√2/2) * 4
= 2√2