The support function of two sets are equal iff the sets are equal

Question

I am not sure how to approach this question from Boyd.

How to show that the support function of two sets $A$ and $B$ are equal iff $A=B$.

The support function for a set $A$ is defined as $S_A(x)=\sup\limits_{y \in A}\langle x,y\rangle$.

For more information on support function go here

@TaraB http://math.stackexchange.com/questions/235926/what-is-a-support-function-sup-z-in-k-langle-z-x-rangle . sup means supremum. — Satvik, Feb 20 '13 at 17:06
OK, so $A$ and $B$ aren't just sets, but ordered sets. How about my other question? — Tara B, Feb 20 '13 at 17:10
Oh, so they're not just ordered sets, but sets with an inner product defined on them! You know you should really state these assumptions, right? — Tara B, Feb 20 '13 at 17:13
@TaraB Sorry I thought that was obvious. The link I provided explains them in detail. — Satvik, Feb 20 '13 at 17:17
No, it is not obvious. If you just say 'set', it means a set with no extra structure. — Tara B, Feb 20 '13 at 17:32

score 0 · Answer 1 · answered Feb 21 '13 at 03:23

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The proposed statement is false. For a counterexample, in the plane, take $A$ to be the unit disk (centered at the origin) and take $B$ to be its boundary, the unit circle. Both have the support function $x\mapsto\Vert x\Vert$. The interior of the disk also has the same support function, as do lots of other sets.

The statement becomes true if you add the assumption that $A$ and $B$ are compact convex sets.

answered Feb 21 '13 at 03:23

Andreas Blass

71,833

Thanks for your answer. Can you tell me how to approach this taking the assumption you mentioned. – Satvik Feb 21 '13 at 07:21

The support function of two sets are equal iff the sets are equal

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