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.

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- Dec 29th 2012, 11:59 AMkezmanConvex bodies Shadow.
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. - Dec 29th 2012, 12:35 PMjakncokeRe: Convex bodies Shadow.
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. - Dec 30th 2012, 08:23 AMkezmanRe: Convex bodies Shadow.
thanks for the help jakncoke !!!!!!

- Jan 25th 2013, 07:38 PMkezmanRe: Convex bodies Shadow.
I re open the thread, cause im missing the inclusion X in S(u,X) knowing that X is open. I have no clue.

Thanks!