Why does taking a convex combination over n co-ordinates result in it being within polygon of n-side

[wispy radish]

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[stoic copper]

It is only within if the polygon is convex itself

[wispy radish]

Assume it is. Then why does it happen?

[stoic copper]

Ok so how do you define mathematically « the interior » of a shape

[wispy radish]

Ohh how 🤔. Perhaps the bounded stuff that has finite area

Is the inside

[stoic copper]

That’s not very mathematical, think about it in terms of boundaries

[wispy radish]

How about "inside is a plane (ABCD) in which every point P that belongs in that plane, has a property that [ABCD]=[PAB]+[PBC]+[PCD]+[PDA]."

[X] represents that area of shape

[wispy radish]

.

Like sum of the areas of all permutations of x,y of [Pxy] = the whole area of the plane

[stoic copper]

Ok so take a polygon. And let’s think of the boundaries (sides). Each side is bounded by 2 vertices right ?

[wispy radish]

Ok

[stoic copper]

So basically, the convex combination of the 2 vertices is all the points in between right ?

[wispy radish]

Yeah

[stoic copper]

Ok so now you see how you can build all the sides just by combinating vertices ?

[wispy radish]

But the thing is when we take a convex combination of two vertices

Their weights make up 1 already

[stoic copper]

Yeah wait for it

Now take any point from the inside and draw a line (any direction). How many many intersections does the line have with the sides ?

[wispy radish]

4

[stoic copper]

2 ?

[wispy radish]

In all for directions rgiht?

Ohh 1 linep

[wispy radish]

Yeah its 2

[stoic copper]

Ok so now these 2 intersections can be combined to find your point inside …

So basically. You can always boil it down to 2 points

[wispy radish]

But what does that mean?

If you have any paper that proves this convex combination over n- veritces

Then please let me know

@stoic copper

[stoic copper]

Alright

[quiet pilot]

@wispy radish How did INMO go?

[wispy radish]

Fucked it

Will not qualify 🫠

[wispy radish]

I prepped geo hard and got played

[wispy radish]

Plus i am very dumb

[quiet pilot]

Did you get any problem correct (even partially)

P3?

[quiet pilot]

Nah dude, You can do it next year.

[wispy radish]

Did P2 and i expect partial on P1

[wispy radish]

Nah

[wispy radish]

How did your inmo go?

[quiet pilot]

That's 20+

[quiet pilot]

only p1

[wispy radish]

Which class you in btw?

[quiet pilot]

10th

[wispy radish]

You got time. Prep hard for the next one

My life is fucked. I was in 11th when i gave this year's inmo. So i have got only attempt + (pressure of iit and boards)

[quiet pilot]

Yet, you aren't starting from scratch. You already reached INMO so it is very possible for you to qual

[quiet pilot]

Yup

[wispy radish]

You migrating ?

[wispy radish]

Are you going to prep for iit ?

[quiet pilot]

Yeah

[quiet pilot]

No

[wispy radish]

Ohh

@stoic copper you there?

[pale karma]

@wispy radish

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