# VC-Dimension of Axis Aligned Right-Angle Triangles and 5-points Convex Hull

I am having trouble proving the following fact about VC dimension of triangles.

Consider right-angle triangles in the plane, with the the right-angle in the lower left corner.

The hypothesis in our case assigns label $$1$$ for points inside the triangle and label $$0$$ otherwise.

Claim is that the VC-dimension of the mentioned above hypothesis class is $$VCdim(\mathcal{H})=4$$. In order to prove it need to show the following:

1. There is a set of cardinality 4 shattered by $$\mathcal{H}$$, and
2. Any set of cardinality greater than 4 is not shattered by $$\mathcal{H}$$. In other words, a set of cardinality $$5$$ cannot be shattered.

The first condition is easy. It suffices to give an example of 4 points in the plane, with any labeling, and show a triangle that covers the labeled points.

I stucked proving the second condition. I split the proof into three cases, which differs by the amount of points that lay inside the convex hull created by those 5 points. The problem is with the case where none of those points lay inside the convex hull (i.e. they create a convex set).

I want to prove that for any combination of those 5 points that lay on a convex hull, there is a labeling that cannot be shattered by a right-angled axis-aligned triangle.

Any help will be appreciated. Thank you.

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