Two-dimensional polytope

RegionPlot[Abs[x]+Abs[y]<=1,{x,-1,1},{y ,-1,1},LabelStyle->(FontSize->14)]

Three-dimensional polytope

RegionPlot3D[Abs[x]+Abs[y]+Abs[z]<=1,{x,-1,1},{y ,-1,1},{z,-1,1},PlotPoints->50,Mesh->None,LabelStyle->(FontSize->14),AxesLabel->{"x","y","z"}]

Two-dimensional quadratic constraint (ellipse)

H=DiagonalMatrix[{1,2}];
c={0,0};
d=-1;
RegionPlot[1/2Dot[{x,y},H.{x,y}]+Dot[c,{x,y}]+d≤0,{x,-2,2},{y ,-2,2},LabelStyle->(FontSize->14)]

Manipulate[H={{1,alpha},{alpha,2}};
c={0,0};
d=-1;
RegionPlot[1/2Dot[{x,y},H.{x,y}]+Dot[c,{x,y}]+d≤0,{x,-2,2},{y ,-2,2},PlotPoints->40,LabelStyle->(FontSize->14)],{{alpha,0},0,1,.05}]

Three-dimensional quadratic constraint (ellipsoid)

H=DiagonalMatrix[{1/4,1/2,1/2}];
c={0,0,0};
d=-1;
RegionPlot3D[1/2Dot[{x,y,z},H.{x,y,z}]+Dot[c,{x,y,z}]+d≤0,{x,-4,4},{y ,-4,4},{z,-4,4},PlotPoints->40,Mesh->None,LabelStyle->(FontSize->14),AxesLabel->{"x","y","z"}]

Manipulate[H=DiagonalMatrix[{alpha,1/2,1/2}];
c={0,0,0};
d=-1;
RegionPlot3D[1/2Dot[{x,y,z},H.{x,y,z}]+Dot[c,{x,y,z}]+d≤0,{x,-4,4},{y ,-4,4},{z,-4,4},PlotPoints->40,Mesh->None,LabelStyle->(FontSize->14),AxesLabel->{"x","y","z"}],{{alpha,1/2},0,1/2,.05}]

H=DiagonalMatrix[{1,2,0}];
c={0,0,-1};
d=0;
RegionPlot3D[1/2Dot[{x,y,z},H.{x,y,z}]+Dot[c,{x,y,z}]+d≤0,{x,-3,3},{y ,-3,3},{z,0,4},PlotPoints->20,Mesh->None,LabelStyle->(FontSize->14),AxesLabel->{"x","y","z"}]

Manipulate[H=DiagonalMatrix[{1,alpha,0}];
c={0,0,-1};
d=0;
RegionPlot3D[1/2Dot[{x,y,z},H.{x,y,z}]+Dot[c,{x,y,z}]+d≤0,{x,-3,3},{y ,-3,3},{z,0,4},PlotPoints->20,Mesh->None,LabelStyle->(FontSize->14),AxesLabel->{"x","y","z"}],{{alpha,2},0,2,.2}]

Two-dimensional ellipse example using shape matrix P

P={{5,-1},{-1,5}}

Eigensystem[P] (* semi-axes *)

RegionPlot[Dot[{x,y},Inverse[P].{x,y}]≤1,{x,-3,3},{y ,-3,3},LabelStyle->(FontSize->14)]