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How to plot dual cone of a given conical domain?

Mathematica Asked on September 26, 2021

If $K$ is a given cone in $mathbb{R}^3$ which passes through origin, the dual cone $K^*$ is defined as $K^*={mathbf{x}inmathbb{R}^3 big| ,, mathbf{x}cdotmathbf{y}ge 0 ,, forall yin K}$.

Given a cone, e.g., $K={(x,y,z)inmathbb{R}^3,,big|,, xy-z^2ge 0}$, how do I plot its dual cone $K^*$, using RegionPlot3d or ContourPlot3d?

2 Answers

The cone K can be plotted as

RegionPlot3D[x y - z^2 >=   0, {x, -1, 1}, {y, -1 , 1}, {z, -1, 1},MaxRecursion -> 5, AxesLabel -> {x, y, z} ]

enter image description here

The dual cone might be plotted in the same we using the additional conditions...

Answered by Ulrich Neumann on September 26, 2021

You are looking for ForAll. As pointed out by Henrik Schumacher, for your particular example $K^*={0}$ since $K=-K$. Thus, I will show how to plot the dual cone one the freely chosen set $xgeq0$, $ygeq0$ and $x+yleq5$.

The function ForAll when given three parameters, ForAll[{x1,...xn},cond,expr] states that expr is true for all xi satisfying the condition cond. Resolve or Reduce then solves such systems. Thus, a MWE looks like

cone = x>=0 && y>=0 && x+y<=5
dual = Resolve[ForAll[{x, y}, cone, px x + py y <= 0], {px, py}]
RegionPlot[cone, {x, -10, 10}, {y, -10, 10}]
RegionPlot[dual, {px, -10, 10}, {py, -10, 10}]

Answered by tommsch on September 26, 2021

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