function [Ax,Ay,b,K] = convexhullConcave(x1,f1,df1,x2,f2,df2,x3,f3,df3,x4,f4,df4,x5,f5,df5)
% x1-x4 corner points, 5 point in center

% Lower bounds from tangents
% z < f(x1) + (x-x1)*df(x1)
% z < f(x2) + (x-x2)*df(x2)
% z < f(x3) + (x-x3)*df(x3)
% z < f(x4) + (x-x4)*df(x4)
% z < f(x5) + (x-x5)*df(x5)

% Possible upper bound from creating hyperplane from three points

f = [f1 f2 f3 f4];
X = [x1 x2 x3 x4];
Z = [X;f];
[n1,d1] = ceiling(Z(:,1),Z(:,2),Z(:,3));
[n2,d2] = ceiling(Z(:,2),Z(:,3),Z(:,4));
[n3,d3] = ceiling(Z(:,3),Z(:,4),Z(:,1));
[n4,d4] = ceiling(Z(:,4),Z(:,3),Z(:,2));
b = [-x1'*df1+f1;
     -x2'*df2+f2;
     -x3'*df3+f3;
     -x4'*df4+f4;
     -x5'*df5+f5];   
Ax = -[df1';df2';df3';df4';df5'];
Ay = [1;1;1;1;1];

% Add a floor 
% n'*z >= d
d1 = min(n1'*Z);
d2 = min(n2'*Z);
d3 = min(n3'*Z);
d4 = min(n4'*Z);
if all(n1'*Z - d1 >= 0)
    b = [b;-d1];
    Ax = [Ax;-n1(1:2)'];
    Ay = [Ay;-n1(3)];
end
if all(n2'*Z - d2 >= 0)
    b = [b;-d2];
    Ax = [Ax;-n2(1:2)'];
    Ay = [Ay;-n2(3)];
end
if all(n3'*Z - d3 >= 0)
    b = [b;-d3];
    Ax = [Ax;-n3(1:2)'];
    Ay = [Ay;-n3(3)];
end
if all(n4'*Z - d4 >= 0)
    b = [b;-d4];
    Ax = [Ax;-n4(1:2)'];
    Ay = [Ay;-n4(3)];
end

function [n,d] = ceiling(p1,p2,p3);
n = cross(p1 - p2, p1 - p3);
n = n/sign(n(3));
d = p1'*n;