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        riddles >> hard >> Minimal triangles in convex figures
        
(Message started by: Deedlit on Jun 5^th, 2005, 3:47pm)

Title: Minimal triangles in convex figures
Post by Deedlit on Jun 5^th, 2005, 3:47pm

Show that, for any convex figure S in the plane, it is possible to choose four points in S such that any three of the four points form a triangle with area at least 1/4 that of S.

Title: Re: Minimal triangles in convex figures
Post by SWF on Jul 1^st, 2005, 7:25pm

It is not difficult to see that it is possible to sandwich the shape between two parallel lines touching the perimeter of shape at points A and B, such that line AB divides the shape into two equal areas.

Let W be the area of the shape. Construct the perpendicular bisector, CD, of AB with points C and D lying on the boundary of the shape. CD goes through point M, the midpoint of AB. Triangles AMC and BCM must have the same area, call it Y=L_AB*L_CM/4.

Since the shape is convex the maximum possible area of the portion of the shape on the C side of AB is L_AB*L_CM, and therefore

L_AB*L_CM >= W/2

Divide by 4 and note left side equals Y:

L_AB*L_CM/4 = Y >= W/8

Triangles AMC and BCM both have an area greater than W/8. By similar reasoning triangles ADM and DBM also have areas greater than W/8.

All four ways of forming a triangle from three of the four points, A, B, C, and D results a triangle made up of two triangles with area greater than or equal to W/8, for a combined area greater than or equal to W/4.

Title: Re: Minimal triangles in convex figures
Post by Deedlit on Aug 2^nd, 2005, 8:46pm

Sigh - another post that I really needed to reply to much earlier.

on 07/01/05 at 19:25:36, SWF wrote:

It's not difficult, but I'm inclined to say that this needs proof.

Quote:

Since the shape is convex the maximum possible area of the portion of the shape on the C side of AB is L_AB*L_CM, and therefore

L_AB*L_CM >= W/2

Unfortunately, this isn't true. Imagine the convex figure to be very wide, and AB to be on a slant. CD is likely to be much shorter than the width of the figure.