# Computation of the Hausdorff Distance between Two Compact Convex Sets
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Set Pair** & \(\mathbf{p}\) & **Method** & **Homotopy** & **Maximum** & **Mean** & **Std** & **Secs** \\ \hline (box, ball \(\cap\) orthant) & 2 & proj grad & false & 1.8284 & 1.3314 & 0.40789 & 0.00163 \\ (box, ball \(\cap\) orthant) & 2 & proj grad & true & 1.8284 & 1.8284 & 0.0 & 0.00349 \\ (box, ball \(\cap\) orthant) & 2 & Frank-Wolfe & false & 0.41421 & 0.16569 & 0.20394 & 0.000564 \\ (box, ball \(\cap\) orthant) & 2 & Frank-Wolfe & true & 0.41421 & 0.41421 & 0.0 & 0.00101 \\ (box, ball \(\cap\) orthant) & 2 & point cloud & false & 1.8159 & 1.8159 & 0.0 & 0.625 \\ (simplex, L1 ball) & 2 & proj grad & false & 1.4142 & 1.4142 & 0.0 & 0.001 \\ (simplex, L1 ball) & 2 & proj grad & true & 1.4142 & 1.4142 & 0.0 & 0.00272 \\ (simplex, L1 ball) & 2 & Frank-Wolfe & false & 1.4142 & 1.4142 & 0.0 & 0.000358 \\ (simplex, L1 ball) & 2 & Frank-Wolfe & true & 1.4142 & 1.4142 & 0.0 & 0.0574 \\ (simplex, L1 ball) & 2 & point cloud & false & 1.4142 & 1.4142 & 0.0 & 0.639 \\ (box, ball \(\cap\) orthant) & 3 & proj grad & false & 2.4641 & 1.6358 & 0.50531 & \(7.34\times 10^{-5}\) \\ (box, ball \(\cap\) orthant) & 3 & proj grad & true & 2.4641 & 2.4641 & 0.0 & 0.000605 \\ (box, ball \(\cap\) orthant) & 3 & Frank-Wolfe & false & 0.73205 & 0.31788 & 0.25266 & \(7.08\times 10^{-5}\) \\ (box, ball \(\cap\) orthant) & 3 & Frank-Wolfe & true & 0.73205 & 0.73205 & 0.0 & 0.000145 \\ (box, ball \(\cap\) orthant) & 3 & point cloud & false & 2.4221 & 2.4221 & 0.0 & 0.63 \\ (simplex, L1 ball) & 3 & proj grad & false & 1.7321 & 1.7321 & 0.0 & \(5.86\times 10^{-6}\) \\ (simplex, L1 ball) & 3 & proj grad & true & 1.7321 & 1.7321 & 0.0 & \(0.000158\) \\ (simplex, L1 ball) & 3 & Frank-Wolfe & false & 1.2247 & 1.2247 & 0.0 & \(4.44\times 10^{-6}\) \\ (simplex, L1 ball) & 3 & point cloud & false & 1.732 & 1.732 & 0.0 & 0.615 \\ (box, ball \(\cap\) orthant) & 10 & proj grad & false & 5.0 & 3.4576 & 0.73165 & 0.0001 \\ (box, ball \(\cap\) orthant) & 10 & proj grad & true & 5.3246 & 5.3246 & 0.0 & 0.000666 \\ (box, ball \(\cap\) orthant) & 10 & Frank-Wolfe & false & 2.0 & 1.2288 & 0.36583 & \(9.71\times 10^{-5}\) \\ (box, ball \(\cap\) orthant) & 10 & Frank-Wolfe & true & 2.1623 & 2.1623 & 0.0 & 0.000183 \\ (box, ball \(\cap\) orthant) & 10 & point cloud & false & 4.8158 & 4.8158 & 0.0 & 0.629 \\ (simplex, L1 ball) & 10 & proj grad & false & 3.1623 & 3.1623 & 0.0 & \(7.66\times 10^{-6}\) \\ (simplex, L1 ball) & 10 & proj grad & true & 3.1623 & 3.1623 & 0.0 & 0.000749 \\ (simplex, L1 ball) & 10 & Frank-Wolfe & false & 2.6667 & 2.6667 & 0.0 & \(7.83\times 10^{-6}\) \\ (simplex, L1 ball) & 10 & Funk-Wolfe & true & 2.6667 & 2.6667 & 0.0 & 0.00196 \\ (simplex, L1 ball) & 10 & point cloud & false & 3.1626 & 3.1626 & 0.0 & 0.613 \\ (box, ball \(\cap\) orthant) & 100 & proj grad & false & 15.248 & 13.016 & 0.69754 & 0.000621 \\ (box, ball \(\cap\) orthant) & 100 & proj grad & true & 19.0 & 19.0 & 0.0 & 0.00126 \\ (box, ball \(\cap\) orthant) & 100 & Frank-Wolfe & false & 7.124 & 6.0078 & 0.34877 & 0.00168 \\ (box, ball \(\cap\) orthant) & 100 & Frank-Wolfe & true & 9.0 & 9.0 & 0.0 & 0.000304 \\ (box, ball \(\cap\) orthant) & 100 & point cloud & false & 15.598 & 15.598 & 0.0 & 0.629 \\ (simplex, L1 ball) & 100 & proj grad & false & 10.0 & 10.0 & 0.0 & \(2.77\times 10^{-5}\) \\ (simplex, L1 ball) & 100 & proj grad & true & 10.0 & 10.0 & 0.0 & \(0.000852\) \\ (simplex, L1 ball) & 100 & Frank-Wolfe & false & 9.8494 & 9.8494 & 0.0 & \(2.21\times 10^{-5}\) \\ (simplex, L1 ball) & 100 & Frank-Wolfe & true & 9.8494 & 9.8494 & 0.0 & \(0.00299\) \\ (simplex, L1 ball) & 100 & proj cloud & false & 9.9493 & 9.9493 & 0.0 & \(0.64\) \\ (box, ball \(\cap\) orthant) & 1000 & proj grad & false & 45.174 & 43.698 & 0.6652 & 0.00591 \\ (box, ball \(\cap\) orthant) & 1000 & proj grad & true & 62.246 & 62.246 & 0.0 & \(0.00875\) \\ (box, ball \(\cap\) orthant) & 1000 & Frank-Wolfe & false & 22.087 & 21.349 & 0.3326 & 0.00256 \\ (box, ball \(\cap\) orthant) & 1000 & Frank-Wolfe & true & 30.623 & 30.623 & 0.0 & \(0.00531\) \\ (box, ball \(\cap\) orthant) & 1000 & point cloud & false & 48.627 & 48.627 & 0.0 & \(1.13\) \\ (simplex, L1 ball) & 1000 & proj grad & false & 31.623 & 31.623 & 0.0 & \(0.000377\) \\ (simplex, L1 ball) & 1000 & proj grad & true & 31.623 & 31.623 & 0.0 & \(0.0113\) \\ (simplex, L1 ball) & 1000 & Frank-Wolfe & false & 31.575 & 31.575 & 0.0 & \(0.000303\) \\ (simplex, L1 ball) & 1000 & Frank-Wolfe & true & 31.575 & 31.575 & 0.0 & \(0.013\) \\ (simplex, L1 ball) & 1000 & point cloud & false & 31.597 & 31.597 & 0.0 & \(1.11\) \\ \hline \hline \end{tabular}
\end{table}