So it’s been about two weeks since I have made my abode here at hyderabad for my Indian Academy of Sciences project. The first week was more or less spent on getting used to the jargon and basic notions of topological dynamical systems, cellular automata, julia sets, the ordinal numbers and mobius groups.
In the course of that week, my guide pointed to me a question he came across: a standard high school problem during a demonstration :
Given a triangle with sides a, b and c constrained by the condition that the perimeter p=a+b+c is fixed, what relationship between a, b and c would lead to a triangle with maximal area?
The interesting thing about this rather simple problem are the multitudinous ways in which the solution can be found. I would not wish to delve into the solution, but a few ways in which this can be done, include calculus, a simple application of the AM-GM inequality, etc.
The result states that the triangle must be equilateral for it to have maximal area meaning that a=b=c=p/3.
Now this is nice, but this immediately puts an excellent inequality in place amongst 3 numbers a, b and c.
Since a, b and c are sides of a triangle they must satisfy the triangle inequality, and the above mentioned question translates into an algebraic inequality for general real numbers satisfying certain conditions.
If a<b+c; b< a+c; and c<a+b then let a+b+c = p and let s = p/2.
Using Heron’s formula for the area of a triangle given the sides a, b and c,
This is a pure algebraic inequality on a, b and c satisfying the three triangle inequalities mentioned above and proving it directly without resort to geometry seems to be a non trivial task to say the least. This is just one out of many such possible examples of what are called geometric inequalities.
The next stage of the question is to try and find out possible such geometric inequalities using similar questions of maximisation and minimisation of geometric functions like area over perimeter, etc.
For example, the next generalisation for the above question could be
For a given quadrilateral a, b, c and d constrained to a fixed perimeter p = a + b + c + d, what is the condition on the four sides such that the area of the quadrilateral thus formed is maximal?
Another interesting question to look at would be,
For a given set of triangles with fixed area, what is the minimal possible perimeter?
It could be an interesting way forward, to try to attempt a direct proof of them same, though I am sure one can use the above (semi)derived geometric inequality to give an easy proof of the problem. Proving that the two problems are equivalent is another very nice problem.
While one avenue is to try and create more geometrical inequalities, the final goal might be to reach the pinnacle of geometric inequalities for any simple closed curve. Something I might write about next time.
Deus ex Machina 