Interior & Boundary Points of Convex Sets in ℝ²

We complete 3 short exercises on interior/boundary points of convex sets in R^2. Given two points belonging to a convex set, which are both interior points, both boundary points, or one of each, we find the possibilities (boundary or interior) for the points on the line segment joining them.

This video is based on an exercise from this book:
Yaglom, I. and Boltyanskii, V., 1961. Convex Figures.

Further reading:

00:00 Intro & Definitions
01:54 Exercise (i)
05:31 Exercise (ii)
07:57 Exercise (iii)
10:56 Conclusion

[guangyao]

thanks Mr Barker! Love your visualisations, was confused after the lecture just now and found your videos first

[mrigayu]

Very intuitive! Great exercise and explanation

[cauchym]

To what kind of spaces can you generalize this result?
Euclidean spaces? Metric spaces? Or even more general?

[Generalth]

Hi Dr. Barker!

I'm starting a math degree this fall, so I've been ramping up my math videos to prepare.

[JoQeZzZ]

What would a proof for the general case (higher dimensions) look like? Does this even work in higher dimensions?

[MrRyanroberson]

Continuijg off of concave/convex is there any usefulness to the set of things which ride the perimeter for some distance? I'm guessing only as a way to divide up the edges of some polygon

[ko-prometheus]

Dude, professor I watch your wonderful lectures!!
But. I am interested in the analysis of a large number of continuous functions, a bundle of functions.
Analysis from the perspective of metaphysics or if you want to call it meta-mathematics.
Are you sure you know this??

[Graknorke]

I know that it clearly is but I'm struggling to justify WHY "everything is a boundary point" is a valid answer, explained in the same terms as the other combinations. you can't just put a boundary point in the middle of the line for example because that just leaves you with "what's between a boundary point and a boundary point" again.

[andreare]

Isn't the fact that the set is convex enough to say that the line segment belongs to the set? It seems to be true by definition.

[firstnamelastname]

Just a small comment : exercise ii e.g. 6:25 you say B belongs to (but you mean is on the boundary of) our set ... of course the reasoning is not affected and is correct

[AngryCoward]

Hi! I'm Convex. Your personal healthcare companion.