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I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3= \{ (x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1\}$
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?

Edit: I would also be interested to know a proof or disproof that the algorithm described in David Speyers community wiki-answer give the function we are looking for in a finite number number of steps (in the limit it does).

Edit: Even though there is an accepted answer, the problem isn't solved. I only accepted the answer, because there was a bounty on the question so I had to accept an incomplete answer.

I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3= \{ (x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1\}$
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?

Edit: I would also be interested to know a proof or disproof that the algorithm described in David Speyers community wiki-answer give the function we are looking for in a finite number number of steps (in the limit it does).

I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3= \{ (x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1\}$
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?

I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3= \{ (x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1\}$
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?

Edit: I would also be interested to know a proof or disproof that the algorithm described in David Speyers community wiki-answer give the function we are looking for in a finite number number of steps (in the limit it does).

I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3=${ $(x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1$}
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?

I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use some analytical tecniques to solve it, but I don't know of any.

A special case of the reduced problem is: Consider the 3-simplex

$\Delta^3= \{ (x_1,x_2,x_3,x_4)|0\leq x_1,x_2,x_3,x_4\leq 1, x_1+x_2+x_3+x_4=1\}$
and functions $f:\Delta^3\to [0,1]$. We want to find the greatest such function satisfying

• $f(½,0,½,0)=f(0,½,0,½)=½$
• If we restrict $f$ to any plane that contains both $(1,0,0,0)$ and $(0,1,0,0)$ we get a convex function
• If we restrict $f$ to any plane that contains both $(0,0,1,0)$ and $(0,0,0,1)$ we get a convex function

Here we say that $f$ is greater than $g$ if $f(x)\geq g(x)$ for all $x\in\Delta^3$. It is clear that there exists a greatest function satisfying the above since it is just the supremum of all functions satisfying the above. In particular, I would like to know if $f$ had to be convex on all of $\Delta^3$ and to know the value of $f(1/4,1/4,1/4,1/4)$.

Is there a known theory for solving this kind of problems?