# [FOM] Theorem on Convex sets

Thu Oct 6 09:18:50 EDT 2005

Here is a proof for N dimensions:

We can assume that the surface of A is made up of "small" (hyper-)
polygons A_i. Project each of these outwards onto the surface of B
(which needn't be convex) along lines parallel to a normal to A_i.
You get shadows B_i on B which do not intersect (convexity of A) and
must be of area not smaller than that of A_i (because we chose
normals to A_i), so the surface area of A must be smaller than that
of B.

Regards,

Jeremy Clark

On Oct 5, 2005, at 9:40 pm, joeshipman at aol.com wrote:

> If A contained in B are convex sets in the plane, the boundary of A is
> no larger than the boundary of B.
>
> Is this true in N dimensions, and if so, who proved it?
>
> In 2 dimensions, I have trouble proving the theorem without rather
> advanced tools. Does anyone know a simple proof?
>
> -- Joe Shipman
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