```% Section 8.2.1, Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 10/09/05
% (a figure is generated)
%
% Given two polyhedra C = {x | A1*x <= b1} and D = {x | A2*x <= b2}, the
% distance between them is the optimal value of the problem:
%           minimize    || x - y ||_2
%               s.t.    A1*x <= b1
%                       A2*y <= b2
% Note: here x is in R^2

% Input data
randn('seed',0);
n = 2;
m = 2*n;
A1 = randn(m,n);
b1 = randn(m,1);
A2 = randn(m,n);
b2 = randn(m,1);

fprintf(1,'Computing the distance between the 2 polyhedra...');
% Solution via CVX
cvx_begin
variables x(n) y(n)
minimize (norm(x - y))
norm(x,1) <= 2;
norm(y-[4;3],inf) <=1;
cvx_end

fprintf(1,'Done! \n');

% Displaying results
disp('------------------------------------------------------------------');
disp('The distance between the 2 polyhedra C and D is: ' );
disp(['dist(C,D) = ' num2str(cvx_optval)]);
disp('The optimal points are: ')
disp('x = '); disp(x);
disp('y = '); disp(y);

%Plotting
figure;
fill([-2; 0; 2; 0],[0;2;0;-2],'b', [3;5;5;3],[2;2;4;4],'r')
axis([-3 6 -3 6])
axis square
hold on;
plot(x(1),x(2),'k.')
plot(y(1),y(2),'k.')
plot([x(1) y(1)],[x(2) y(2)])
title('Euclidean distance between 2 polyhedron in R^2');
xlabel('x_1');
ylabel('x_2');
```
```Computing the distance between the 2 polyhedra...
Calling Mosek 9.1.9: 15 variables, 5 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
Name                   :
Objective sense        : min
Type                   : CONIC (conic optimization problem)
Constraints            : 5
Cones                  : 5
Scalar variables       : 15
Matrix variables       : 0
Integer variables      : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00
Lin. dep.  - tries                  : 1                 time                   : 0.00
Lin. dep.  - number                 : 0
Presolve terminated. Time: 0.00
Problem
Name                   :
Objective sense        : min
Type                   : CONIC (conic optimization problem)
Constraints            : 5
Cones                  : 5
Scalar variables       : 15
Matrix variables       : 0
Integer variables      : 0

Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 3
Optimizer  - Cones                  : 3
Optimizer  - Scalar variables       : 10                conic                  : 7
Optimizer  - Semi-definite variables: 0                 scalarized             : 0
Factor     - setup time             : 0.00              dense det. time        : 0.00
Factor     - ML order time          : 0.00              GP order time          : 0.00
Factor     - nonzeros before factor : 6                 after factor           : 6
Factor     - dense dim.             : 0                 flops                  : 5.40e+01
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME
0   4.0e+00  1.0e+00  2.0e+00  0.00e+00   1.000000000e+00   0.000000000e+00   1.0e+00  0.00
1   1.6e+00  3.9e-01  7.4e-01  -7.69e-02  1.745839661e+00   1.452430446e+00   3.9e-01  0.02
2   4.4e-01  1.1e-01  9.0e-02  9.65e-01   2.438967963e+00   2.285065173e+00   1.1e-01  0.02
3   4.9e-02  1.2e-02  3.6e-03  9.07e-01   2.134713080e+00   2.118964344e+00   1.2e-02  0.02
4   3.7e-04  9.4e-05  2.4e-06  9.88e-01   2.121484330e+00   2.121364392e+00   9.4e-05  0.02
5   2.2e-05  5.5e-06  3.4e-08  1.00e+00   2.121333459e+00   2.121326455e+00   5.5e-06  0.02
6   1.1e-06  2.7e-07  3.6e-10  1.00e+00   2.121321075e+00   2.121320737e+00   2.7e-07  0.02
7   2.1e-08  5.2e-09  1.0e-12  1.00e+00   2.121320359e+00   2.121320352e+00   5.2e-09  0.02
Optimizer terminated. Time: 0.02

Interior-point solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: 2.1213203591e+00    nrm: 4e+00    Viol.  con: 8e-13    var: 0e+00    cones: 1e-08
Dual.    obj: 2.1213203524e+00    nrm: 1e+00    Viol.  con: 0e+00    var: 1e-12    cones: 4e-09
Optimizer summary
Optimizer                 -                        time: 0.02
Interior-point          - iterations : 7         time: 0.02
Basis identification  -                        time: 0.00
Primal              - iterations : 0         time: 0.00
Dual                - iterations : 0         time: 0.00
Clean primal        - iterations : 0         time: 0.00
Clean dual          - iterations : 0         time: 0.00
Simplex                 -                        time: 0.00
Primal simplex        - iterations : 0         time: 0.00
Dual simplex          - iterations : 0         time: 0.00
Mixed integer           - relaxations: 0         time: 0.00

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +2.12132

Done!
------------------------------------------------------------------
The distance between the 2 polyhedra C and D is:
dist(C,D) = 2.1213
The optimal points are:
x =
1.5000
0.5000

y =
3.0000
2.0000

```