# Plotting a $3$-dimensional convex shape consisting of portions of six spheres

How can we plot a convex shape $$C$$ in the $$3$$-dimensional Euclidean space, whose boundary $$\partial C$$ is formed by portions of six different spheres, one portion per sphere? The radius of each sphere is the same, $$a+1$$, and the coordinates of six spheres centers are $$(a,0,0), (-a,0,0), (0,a,0), (0,-a,0), (0,0,a), (0,0,-a)$$, where the parameter $$a$$ is a non negative real number. Hence, each sphere passes through one and only one of the following points for $$a>0$$:

$$(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)$$.

Finally, for each of such points $$\mathbf{p}$$, the portion of the sphere $$S_{\mathbf{p}}$$ passing through it consists of all points of $$S_{\mathbf{p}}$$ whose distance from $$\mathbf{p}$$ is smaller than the distance between $$\mathbf{p}$$ and any point of the other five spheres.

Note that for $$a=0$$, $$C$$ is the sphere with unit radius centered at the origin. As $$a$$ grows we have a shape always lying in $$[-1,1]^3$$, where the Gaussian curvature is the same at any point of $$C$$ is everywhere the same (and larger than the unit sphere as $$a$$ grows). For $$a\to\infty$$, we have the cube $$[-1,1]^3$$. Hence, this convex shape can be viewed as a continuous transformation from the unit sphere to its bounding box $$[-1,1]^3$$.

Using the new CSGRegion functionality, we can an extremely fast and accurate result:

Manipulate[
CSGRegion["Intersection",
Ball[a #, a + 1] & /@ {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1,
0}, {0, 0, 1}, {0, 0, -1}}],
{a, 0, 5}
] A somewhat slow solution is using an ImplicitRegion:

Manipulate[
sp = Ball[#, 1 + a] & /@ {{a, 0, 0}, {\[Minus]a, 0, 0}, {0, a,
0}, {0, \[Minus]a, 0}, {0, 0, a}, {0, 0, \[Minus]a}};
Region[ImplicitRegion[
And @@ ({x, y, z} \[Element] # & /@ sp), {x, y, z}]]
, {{a, .5}, 0, 1}] 