(Update)
My current result is $82$ points:
consider this point set:
pts = {
{39331, -1787, 125739},
{-42020, -78476, 96709},
{97017, -83209, 30835},
{-17033, 70737, 109597},
{-54599, 29504, 115688},
{-69547, 63866, 91701},
{-84862, -62280, -80052},
{111630, -49662, -51118},
{110858, 44843, -58218},
{7570, -94324, 91248},
{115828, -36578, 50910},
{-103422, 33617, 73525},
{13903, 130088, -24865},
{-48488, -30540, -119577},
{13546, 105208, 78574},
{92754, -90941, -22055},
{-87842, -12726, -97961},
{17890, -95311, -90222},
{-32617, 127358, -17688},
{-83770, -100939, 6478},
{-67513, -103415, -46172},
{-15435, 70574, -111233},
{42948, 122369, 28253},
{82827, -31757, -98975},
{-8841, 14824, 130515},
{-31918, -116156, 52485},
{-124638, 33189, 26548},
{46151, -58101, 108697},
{-107711, 76927, -3256},
{8590, -131155, -3832},
{-2349, -45047, 123671},
{-67052, 113066, 17470},
{-49845, -26471, 118738},
{45038, -56580, -110986},
{124167, -45279, 903},
{60780, -115738, 12319},
{-109374, -68092, -27125},
{-40207, -124921, 2722},
{74952, 40665, 100449},
{88162, -58830, 78010},
{60461, 114907, -29946},
{110136, -3355, -73936},
{70896, 79060, -79787},
{56554, -97875, 67358},
{72446, -84584, -71147},
{30586, 57713, 114256},
{-15936, -120088, -52161},
{-480, -46761, -124154},
{-72908, 103917, -38653},
{-101424, 28721, -80454},
{-45115, 103290, 68859},
{41881, -117921, -41667},
{-74575, -93889, 53049},
{108114, 53390, 54482},
{15266, -123265, 42434},
{40723, -3854, -126221},
{90334, 94409, 22158},
{96396, 85431, -32579},
{-63349, 75478, -88497},
{122169, 52183, -1811},
{108487, 5280, 74810},
{-88785, -956, 96779},
{-7851, 14221, -131625},
{64857, 88850, 73124},
{23713, 102177, -81511},
{129972, 1413, -27143},
{-119337, -14421, 52312},
{-88103, -51438, 82718},
{-10887, 127563, 33645},
{33805, 54367, -116181},
{-102814, 64657, -52366},
{-126644, 25744, -26822},
{-25275, 110536, -68979},
{-112785, -59627, 30034},
{-129858, -19908, 289},
{-36740, -84005, -95750},
{78058, 29755, -103069},
{-118373, -22382, -53597},
{-55526, 28946, -116699},
{-94065, 79056, 48080},
{80742, -15619, 102763},
{129505, 8123, 26059}
}
Then (Mathematica code)
Volume[ConvexHullMesh[pts]]
is $\approx 9.00744\times10^{15}$.
And Mathematica sketch:
ConvexHullMesh[pts]]
Another picture. If all vertices of a face are at distance one from another vertex, the face is colored blue.
Since all point coordinates are integer, then one can write it directly (with arbitrary small computational errors):
$$Diameter = \sqrt{68\;719\;348\;253} \approx 262\;143.\;754\;938;$$
$$Volume = \dfrac{54\;044\;635\;971\;533\;362}{6} \approx 9\;007\;439\;328\;588\;893.\;666\;667.$$
If multiply all coordinates by $\dfrac{1}{2^{18}}$, then we'll get:
$$Diameter = \frac{\sqrt{68\;719\;348\;253}}{262\;144} \approx 0.999\;999\;065;$$
$$Volume = \dfrac{54\;044\;635\;971\;533\;362}{2^{54}\times 6} \approx 0.\;500\;013\;326.$$
Note: when add any point (with real coordinates) rather close to (the center of) any face, one will get the set of $83, 84, ...$ points with described property.
Best Answer
If any three vertices (defined by vectors $A$, $B$, $C$) form a triangle as a part of the skin of a volume, then the volume of the triangle pyramid is formed by the three vertices, and the origin is
$${\rm volume}(A,B,C) = \frac{ A \cdot ( B \times C) }{6}$$
Here $\cdot$ is the vector dot product, and $\times$ the vector cross product. The above is often called the vector triple product.
To find the total volume, split the object into triangles and add up all the volumes from above. If a face normal is away from the origin it will count as positive volume, and towards the origin, it will be negative. This way due to Green's theorem in total only the volume enclosed by the mesh will be counted.
Please also note the area of the triangle is
$${\rm area}(A,B,C)=\tfrac{1}{2}\|A\times B+B\times C+C\times A\|$$
Here is a simple example with
$$ \begin{aligned}{A} & =\begin{pmatrix}40\\ 0\\ 0 \end{pmatrix} & {B} & =\begin{pmatrix}0\\ 25\\ 0 \end{pmatrix} & {C} & =\begin{pmatrix}0\\ 0\\ 8 \end{pmatrix}\end{aligned}$$
The volume is
$$ V=\tfrac{1}{6}\begin{pmatrix}40\\ 0\\ 0 \end{pmatrix}\cdot\left(\begin{pmatrix}0\\ 25\\ 0 \end{pmatrix}\times\begin{pmatrix}0\\ 0\\ 8 \end{pmatrix}\right)=\tfrac{1}{6}\begin{pmatrix}40\\ 0\\ 0 \end{pmatrix}\cdot\begin{pmatrix}200\\ 0\\ 0 \end{pmatrix}=\frac{8000}{6}=1333.3\overline{3}\;\checkmark$$
Which is the value that matches with CAD.
And here is a more complex example
which I matched with CAD also. The STL file contained 5088 vertices and 1696 triangular faces. The CAD model gave me a volume of
46085.4
and a Fortran program I checked with gave me the value46086.1
Now granted the Fortran program was intended to derive the mass properties from an STL file, such as volume, the center of mass, and mass moment of inertia tensor.
In the code there is clearly visible the line
i_V = dot(A, cross(B, C))/6
which computes the volume for each triangle.For completeness here is the processing code