(*
{0,0,0} - starting point;
1 - minimal distance;
{a,0,b} - point generating antiprism;
{x,y,z} - vector from {a,0,b} pointing to another vertex of antiprism centered at {a,0,b}.
*)
NMaximize[{
Min[
EuclideanDistance[{-a,0,b},{x+a,y,z+b}],
EuclideanDistance[{0,a,b},{x+a,y,z+b}],
EuclideanDistance[{0,-a,b},{x+a,y,z+b}],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}],

EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],

EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]
],
{a^2+b^2==1, (* Condition for distance 1 *)
a^2>=b^2, (* Condition for side of the base to be at least 1 *)
a^2(1-Sqrt[2])+3b^2>=0, (* Condition for distance between different bases at least 1 *)
x^2+y^2+z^2==1, (* Condition for the antiprism at {a,0,b} *)
a*x+b*z==a^2-b^2 (* Condition for the second antiprism to be congruent *)
}
},
{x,y,z,a,b}]
{0.598038,{x->0.373229,y->0.768033,z->0.520409,a->-0.73451,b->0.678598}}
(*
{0,0,0} - starting point;
1 - minimal distance;
{a,0,b} - point generating antiprism;
{x,y,z} - vector from {a,0,b} pointing to another vertex of antiprism centered at {a,0,b}.
*)
NMaximize[{
Min[
EuclideanDistance[{-a,0,b},{x+a,y,z+b}],
EuclideanDistance[{0,a,b},{x+a,y,z+b}],
EuclideanDistance[{0,-a,b},{x+a,y,z+b}],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}],

EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)],

EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],

EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)],
EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]

],
{a^2+b^2==1, (* Condition for distance 1 *)
a^2>=b^2, (* Condition for side of the base to be at least 1 *)
a^2(1-Sqrt[2])+3b^2>=0, (* Condition for distance between different bases at least 1 *)
x^2+y^2+z^2==1, (* Condition for the antiprism at {a,0,b} *)
a*x+b*z==a^2-b^2 (* Condition for the second antiprism to be congruent *)
}
},
{x,y,z,a,b}]
{0.598038,{x->0.373229,y->0.768033,z->0.520409,a->-0.73451,b->0.678598}}
(*
{0,0,0} - starting point;
1 - minimal distance;
{a,0,b} - point generating antiprism;
{x,y,z} - vector from {a,0,b} pointing to another vertex of antiprism centered at {a,0,b}.
*)
f[x_]:=Piecewise[{{3,x<0.01},{x,x>=0.01}}];
(* Function to make sure that minimal distance between different points*)

NMaximize[{
Min[
f[EuclideanDistance[{-a,0,b},{x+a,y,z+b}]],
f[EuclideanDistance[{0,a,b},{x+a,y,z+b}]],
f[EuclideanDistance[{0,-a,b},{x+a,y,z+b}]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{x+a,y,z+b}]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{x+a,y,z+b}]],

f[EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}+{b*y,z*a-x*b,-a*y}/(2*b)]],

f[EuclideanDistance[{-a,0,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{0,a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{0,-a,b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(x+a)/2,y/2,(z+b)/2}-{b*y,z*a-x*b,-a*y}/(2*b)]],

f[EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],

f[EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]+{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],

f[EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}+{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],

f[EuclideanDistance[{-a,0,b},
{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{0,-a,b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]],
f[EuclideanDistance[{-a/Sqrt[2],-a/Sqrt[2],b},{(3*a-x)/2,-y/2,(3*b-z)/2}-{(x+a)/2,y/2,(z+b)/2}/Sqrt[2]-{b*y,z*a-x*b,-a*y}/(2*Sqrt[2]*b)]]

],
{a^2+b^2==1, (* Condition for distance 1 *)
a^2>=b^2, (* Condition for side of the base to be at least 1 *)
a^2(1-Sqrt[2])+3b^2>=0, (* Condition for distance between different bases at least 1 *)
x^2+y^2+z^2==1, (* Condition for the antiprism at {a,0,b} *)
a*x+b*z==a^2-b^2 (* Condition for the second antiprism to be congruent *)
}
},
{x,y,z,a,b}]


{0.598038,{x->0.373229,y->0.768033,z->0.520409,a->-0.73451,b->0.678598}}

NMaximize[{
Min[EuclideanDistance[{x,y,1-b},{a,0,b}],
EuclideanDistance[{y,-x,1-b},{a,0,b}],
EuclideanDistance[{-x,-y,1-b},{a,0,b}],
EuclideanDistance[{-y,x,1-b},{a,0,b}]]-Sqrt[a^2+b^2],
{b>0,
b<0.5,
a^2>=b^2,
a^2+b^2>1,
a^2(1-Sqrt[2])+3b^2>=0,
x^2+y^2== a^2,
x>0,
y>0,
a>0
}
},
{x,y,a,b}]
{-0.221054,{x->0.662827,y->0.662827,a->0.937379,b->0.348311}}
Maximize[{
RankedMin[{EuclideanDistance[{x,y,1-b},{a,0,b}],
EuclideanDistance[{y,-x,1-b},{a,0,b}],
EuclideanDistance[{-x,-y,1-b},{a,0,b}],
EuclideanDistance[{-y,x,1-b},{a,0,b}]},1]+
RankedMin[{EuclideanDistance[{x,y,1-b},{a,0,b}],
EuclideanDistance[{y,-x,1-b},{a,0,b}],
EuclideanDistance[{-x,-y,1-b},{a,0,b}],
EuclideanDistance[{-y,x,1-b},{a,0,b}]},2]-Sqrt[a^2+b^2]-1,
{b>0,
b<0.5,
a^2>=b^2,
a^2+b^2>1,
a^2(1-Sqrt[2])+3b^2>=0,
x^2+y^2== a^2,
x>0,
y>0,
a>0
}
},
{x,y,a,b}]
{-0.336696,{x->0.937379,y->0.,a->0.937379,b->0.348311}}
Maximize[{EuclideanDistance[{x,y,1-b},{a,0,b}]+EuclideanDistance[{y,-x,1-b},{a,0,b}]-Sqrt[a^2+b^2]-1,
{b>0,
b<0.5,
a^2>=b^2,
a^2+b^2>1,
a^2(1-Sqrt[2])+3b^2>=0,
x^2+y^2== a^2,
x>=0,
y>=0,
a>=0
}
},
{x,y,a,b}]
{-0.336696,{x->0.937379,y->0.,a->0.937379,b->0.348311}}