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.


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...

  • $\begingroup$ thanks for the term $\endgroup$ – Beverlie Aug 21 '17 at 13:16
  • $\begingroup$ 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. $\endgroup$ – kimchi lover Aug 21 '17 at 23:49
  • $\begingroup$ ... which is the most well-introduced course or book for understanding your brief mention? $\endgroup$ – Beverlie Aug 22 '17 at 5:07
  • $\begingroup$ 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.) $\endgroup$ – kimchi lover Aug 22 '17 at 13:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.