Plotting maxima within a simplex

Question

I am trying to draw a particular maximum plot within a simplex. I have functions based on two variables:

Write p,q for the two variables where p,q \geq 0 and p + q \leq 1. This defines the simplex I am interested in. (Think of, say, p on the x-axis and q on the y-axis.)

The functions are then of the following form:

f(p,q) = 22.5p + 10(1-p)

g(p,q) = 40(1-p-q) + 15(p+q)

h(p,q) = 10(1-q) + 22.5q.

(I am giving examples of functions---there are several I want to look at.)

I am interested in which function is "highest" for any given (p,q). So, for any given (p,q) in the simplex, I want to look at the max {f, g, h}. (The functions are such that almost always the maximum will be unique.) I then want it plot the point (p,q) as, say, red if f is the maximum function, blue if g is the maximum function, and green if h is the maximum function.

Does anyone have any idea how this can be done within mathematica?

Answer 1

Piggybacking on @kglr's answer, but using a right triangle rather than equilateral, so we can see p and q on the axes and to highlight the symmetry between f and h.

How about:

Plot3D[max[p, q], {p, q} \[Element] Triangle[{{0, 0}, {1, 0}, {0, 1}}],
  ColorFunction ->
  (Which[#3 == f[#1, #2], Red, #3 == g[#1, #2], Green, #3 == h[#1, #2], Blue] &),
  ColorFunctionScaling -> False, PlotPoints -> 100]

A dirty trick to get a 2D plot is to move the ViewPoint to {0, 0, \[Infinity]}:

Plot3D[max[p, q], {p, q} \[Element] 
  Triangle[{{0, 0}, {1, 0}, {0, 1}}], 
 ColorFunction -> (Which[#3 == f[#1, #2], Red, #3 == g[#1, #2], 
     Green, #3 == h[#1, #2], Blue] &), ColorFunctionScaling -> False, 
 PlotPoints -> 100, ViewPoint -> {0, 0, \[Infinity]}, 
 Axes -> {True, True, False}, Mesh -> None]

Here's another possible 2D solution:

DensityPlot[
  Which[
    max[p, q] == f[p, q], 1,
    max[p, q] == g[p, q], 2,
    max[p, q] == h[p, q], 3
  ],
 {p, q} \[Element] Triangle[{{0, 0}, {1, 0}, {0, 1}}], 
 PlotPoints -> 100, 
 ColorFunction -> (Which[#1 == 1, Red, #1 == 2, Green, #1 == 3, Blue] &),
 ColorFunctionScaling -> False
]

(similar output)

Of course hard coding the number of functions is ugly; it'd be nice to have this accept an arbitrary number of functions to compare!

Answer 2

ClearAll[f, g, h]

f[p_, q_] := 22.5 p + 10 (1 - p)
g[p_, q_] := 40 (1 - p - q) + 15 (p + q)
h[p_, q_] := 10 (1 - q) + 22.5 q

max[p_, q_] := Max[f[p, q], g[p, q], h[p, q]]


Plot3D[max[p, q], {p, q} ∈ SSSTriangle[1, 1, 1], 
  Filling -> Bottom, Exclusions -> None]

{rf, rg, rh} =  Quiet @
  Reduce[#[p, q] >= max[p, q] , {p, q} ∈ SSSTriangle[1, 1, 1]] & /@ {f, g, h};

RegionPlot[{rf, rg, rh}, {p, 0, 1}, {q, 0, 1}, PlotStyle -> {Red, Blue, Green}]