In[1]:= NMinValue[z, {x, y, z} \[Element] Tetrahedron[]]

Out[1]= -0.204124

In[2]:= NMinValue[z, {x, y, z} \[Element] Tetrahedron[], 
 Method -> "SimulatedAnnealing"]

During evaluation of In[2]:= NMinValue::cvdiv: Failed to converge to a solution. The function may be unbounded.

Out[2]= -7.28737*10^104

In[7]:= RegionMember[Octahedron[], {x, y, z}]

Out[7]= RegionMember[Octahedron[], {x, y, z}]

In[13]:= FindMinimum[{z, RegionMember[#, {x, y, z}]}, {x, y, 
    z}] & /@ {Tetrahedron[], Cube[], Dodecahedron[], Icosahedron[]}

During evaluation of In[13]:= FindMinimum::dinfeas: The dual is infeasible, which implies that the primal optimization is either unbounded or infeasible.

During evaluation of In[13]:= FindMinimum::ubnd: The problem is unbounded.

During evaluation of In[13]:= FindMinimum::dinfeas: The dual is infeasible, which implies that the primal optimization is either unbounded or infeasible.

During evaluation of In[13]:= FindMinimum::dinfeas: The dual is infeasible, which implies that the primal optimization is either unbounded or infeasible.

During evaluation of In[13]:= General::stop: Further output of FindMinimum::dinfeas will be suppressed during this calculation.

Out[13]= {{-\[Infinity], {x -> Indeterminate, y -> Indeterminate, 
   z -> Indeterminate}}, {-\[Infinity], {x -> Indeterminate, 
   y -> Indeterminate, 
   z -> Indeterminate}}, {-\[Infinity], {x -> Indeterminate, 
   y -> Indeterminate, 
   z -> Indeterminate}}, {-\[Infinity], {x -> Indeterminate, 
   y -> Indeterminate, z -> Indeterminate}}}

In[1]:= Minimize[z, {x, y, z} \[Element] Cube[]]

During evaluation of In[1]:= Minimize::natt: The minimum is not attained at any point satisfying the given constraints.

Out[1]= {-\[Infinity], {x -> Indeterminate, y -> Indeterminate, 
  z -> Indeterminate}}

In[2]:= Minimize[z, {x, y, z} \[Element] Sphere[]]

Out[2]= {-1, {x -> 0, y -> 0, z -> -1}}

There are other problems

In[1]:= t = Tetrahedron[]

Out[1]= Tetrahedron[]

In[2]:= Volume[t]

Out[2]= 1/(6 Sqrt[2])

In[3]:= Minimize[z, {x, y, z} \[Element] t]

During evaluation of In[3]:= Minimize::natt: The minimum is not attained at any point satisfying the given constraints.

Out[3]= {-\[Infinity], {x -> Indeterminate, y -> Indeterminate, 
  z -> Indeterminate}}

In[4]:= trm = RegionMember[t, {x, y, z}]

Out[4]= (x | y | z) \[Element] Reals && 1/(2 Sqrt[6]) + z <= 0 && 
 1/(2 Sqrt[6]) - 2 Sqrt[2] (1/(2 Sqrt[3]) + x) + z <= 0 && 
 1/(2 Sqrt[6]) + Sqrt[2] (-(1/Sqrt[3]) + x) - Sqrt[6] y + z <= 
  0 && -Sqrt[(2/3)] + 1/(2 Sqrt[6]) + Sqrt[2] x - 
   2 Sqrt[3] (-(1/(3 Sqrt[2])) - Sqrt[2]/3) y + z <= 0

In[5]:= trmfs = 
 FullSimplify[trm, Assumptions -> (x | y | z) \[Element] Reals]

Out[5]= Sqrt[6] + 12 z <= 0 && 4 z <= Sqrt[6] + 8 Sqrt[2] x && 
 Sqrt[2] x + z <= 1/2 Sqrt[3/2] (1 + 4 y) && 
 4 (Sqrt[2] x + Sqrt[6] y + z) <= Sqrt[6]

In[6]:= Volume[ImplicitRegion[trmfs, {x, y, z}]]

Out[6]= \[Infinity]