Direction of Movement of the Element of Minimal Norm in a Moving Convex Set

We show that if

is a nonempty closed convex subset of a real Hilbert space

H H

,

e e

is a non-zero arbitrary vector in

H H

and for each

t\in \mathbb{R} t ∈ R

,

z(t) z ( t )

is the closest point in

K + te K + t e

to the origin, then the angle

z(t) z ( t )

makes with

e e

is a decreasing function of

t t

while

z(t)\neq 0 z ( t ) ≠ 0

, and the inner product of

z(t) z ( t )

with

e e

is increasing.