# Least Required Number for Constructing n-dimensional Convex Hull

Just had learn the concept of Convex set and Convex Hull.

At this point I had figured of my self regarding the question as following:

"Do I always need to have at least $n+1$ number of elements to construct a convex hull which is embedded in $\Bbb R^n$ space?"

This question is not fully elaborated with a commonly accepted terminologies because of lack of my experience in mathematics so will plot more examples to evade elusiveness.

First, If I have 2-dimensional space, which is correspondent to a sheet of paper , I need at least 3 different elements to construct a polygon which requires at least 2-dimensional space to be fully embedded in it(i.e. triangle would be convex hull which is fully embedded into the 2-dimensional plane with least number of elements).

Also, in a similar vein, If I have 3-dimensional space, which is correspondent to $\Bbb R^3$, I need at least 4 elements or points to construct a 3-dimensional object.

Upon this sense I would like to generalize this idea or notion into n-dimensional sense.

However, where do I have to start from to deal with more or equal to 4-dimensional space which is not visualizable to prove this generalization?

Any related concept or already-existing theorem or statement would be also appreciated.

## 1 Answer

There is such a thing as a simplex, a generalization of triangles, tetrahedra, and so on. The $n$-simplex is the convex hull of $n+1$ points in $\mathbb R^n$. There's a lot more to say about them, but...

• thanks for the term – Beverlie Aug 21 '17 at 13:16
• Let me mention one fact about simplices: for any bounded polytope $P$ in $m$ space there is a simplex $S$ in some $n$ space, such that $P$ is the image of $S$ under an affine map. – kimchi lover Aug 21 '17 at 23:49
• ... which is the most well-introduced course or book for understanding your brief mention? – Beverlie Aug 22 '17 at 5:07
• From this and that I assume you are a high school student or a university student just beginning to study mathematics? Maybe the Convex Figures And Polyhedra by iusternik is pitched right for you. There is a book Convex Polytopes by Grünbaum, the easier chapters might be accessible to you. My library has a book Convex Sets by Valentine, which looks like a solid undergraduate treatment. (I apologize if I've misjudged your level. It is hard to make sensible book recommendations without knowing something about the reader.) – kimchi lover Aug 22 '17 at 13:41