Since is an upper semicontinuous function, hence it attains its minimum in any closed bounded interval of.
Since (varphi "'(x)) is continuous on ([alpha, beta ]), it attains its minimum and maximum value on ([alpha, beta ]), i.e. there exist (m=min_{xin [alpha,beta ]} varphi "'(x)) and (M=max_{xin [alpha,beta ]}varphi "'(x)).
There is even a lower setting for the smoothness factor since A n attains its minimum norm when it is purely imaginary.
Indeed, it has been proved, in [1], that f = ⦀ A ν X B 1 − ν + A 1 − ν X B ν ⦀ is a convex function of ν on [ 0, 1 ] with symmetry about ν = 1 2, and attains its minimum there and it has a maximum at ν = 0 and ν = 1.
Obviously, attains its minimum at.
Moreover, is weakly lower-semicontinuous in, and hence attains its minimum in.
The situation changes when if the convex function attains its minimum nonuniquely.
Therefore, ϕ ( t ) is convex on [ 0, 1 ] and attains its minimum at t = 1 2. □.
is convex on [0,2] and attains its minimum at r = 1.
is a continuous convex function on [0,1] and attains its minimum at.
As follows by calculus, in each interval, attains its minimum at the endpoints.
