For
and
we set
is the shadow of X when the light is coming from the direction u.
Let
.Prove .
(a) If X is a convex subset of
, then is also convex.
(b) If X is open in
, then is open and .
Im not sure how to aproach both of them.
Thanks.
For
and
we set
is the shadow of X when the light is coming from the direction u.
Let
.Prove .
(a) If X is a convex subset of
, then is also convex.
(b) If X is open in
, then is open and .
Im not sure how to aproach both of them.
Thanks.
Well for
For
a) Any element of S(u;X) has the form Well, to show convexity you have to show that for any So multiplying the p through we get since X is convex, . So we got , then factor out the t to get , which is which is since
b)
Assume X is open.
To show that S(u;X) is open, for any element i need to show that there exists a positive radius r such that this implication holds, if then p must be in S(u;X).
Now, since j is of the form where . Since X is open, there exists a radius such that this implication holds, if then .
Now i will show that for the radius derived from j = tu + x (it is the radius for x, an element of X ), if then .
Now, assume So . So this means that which means Thus , thus which means that S(u;X) is open.