Volume of 1/2 using hull of finite point set with diameter 1
(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.
The current best known volume bound by 97 points is .4999609.
Currently, 98 points is the lower bound, with a current best volume of .5001309. Here's a simplified version of 98 points bounding a volume of 1/2.
pts =
({{25161,15239,15738}, {-19559,26175,-6775}, {-4006,24157,-22572},
{26243,20473,1977}, {3626,21357,25058}, {-16176,-20481,19867},
{19600,24155,-12113}, {-20547,-23666,-10570}, {15495,-4185,28992},
{-30124,-3246,14209}, {20518,-25644,-4873}, {-27314,-16747,8407},
{24128,-1894,-22531}, {28918,-5278,15688}, {-9651,-30430,-8794},
{32124,-8719,2667}, {7,-28881,15714}, {-27983,17284,5117},
{31307,-5904,-8466}, {-29112,-13355,-7321}, {-11197,31241,2423},
{-16493,-6895,-28142}, {-4835,-19067,26682}, {-27426,9201,16379},
{-14201,11757,-27212}, {-13743,26726,14584}, {15279,29455,544},
{23529,14005,-19238}, {3316,13250,-29746}, {32829,6364,-1267},
{-17854,20459,-18567}, {23584,2651,22825}, {-31684,1388,-8352},
{-27907,13905,-10730}, {-20649,-25299,3031}, {-7914,13803,28914},
{-5685,31345,-9492}, {8966,26190,-18664}, {13899,26052,14660},
{-19436,-5349,26308}, {12582,-26252,15883}, {9,3354,32858},
{15269,12403,25837}, {-27834,-4988,-17084}, {-32368,2810,4363},
{-18997,10095,25337}, {2653,-27021,-19141}, {14688,14913,-25846},
{31456,5955,8605}, {22783,-19010,-14896}, {7504,-30451,-9382},
{11921,-30012,5350}, {-6237,3312,-32379}, {5452,32220,3112},
{4892,30443,-11704}, {20633,-13337,22304}, {13320,-19143,-23609},
{-15014,-19141,-22978}, {1043,-33106,290}, {26794,-18693,4769},
{-22610,4416,-23628}, {-11418,-29433,9363}, {-1517,-12798,-30638},
{23056,-19104,13931}, {10706,-3817,-31400}, {7894,7519,30955},
{5255,-14276,29698}, {-1278,31511,10073}, {-9977,-7220,30694},
{-6921,22257,22989}, {29137,11121,-9805}, {-31068,-7994,1171},
{-18843,17399,20269}, {-8660,-25176,-19425}, {12922,-18519,24195},
{27836,-15187,-6057}, {-21640,22315,9878}, {-6851,-32147,3772},
{-23691,-11945,18070}, {4488,2283,-32516}, {14245,5686,-28818},
{-22214,-12745,-19863}, {22760,-11594,-20279}, {-12368,2766,29586},
{5081,28759,15175}, {4609,-19768,-25959}, {-1159,-8679,31850},
{13750,-25002,-14800}, {-24581,10464,-18734}, {4255,-22632,22566},
{28756,3237,-13827}, {-9454,-14279,-28365}, {19317,-24852,6639},
{12130,-11039,-28322}, {-6116,-4669,-31895}, {-30516,9944,-2126},
{-24217,1636,21574}, {20094,24320,7885}} * 1/(2^20 +2))//N;
Max[EuclideanDistance[#[[1]], #[[2]]] & /@ Subsets[pts, {2}]]
0.999997
Volume[ConvexHullMesh[N[pts]]]
0.500044
Here's a picture of a non-simplified version of the 98 points. All three vertices of a blue triangle are at distance 1 from a vertex on the other side.
A similar picture for 36 points, which bounds a volume of 0.4699687. I believe this is optimal. Some believed-optimal solutions for 7, 8, 9, 10, 11, and 16 points are at my blog article and at the demo biggest little polyhedron, which has been updated.
A related question is to ask the volume of the largest $n$-vertex polyhedron which can be inscribed in a sphere of given size. It is important to recognize that this is a different problem, as a polyhedron of diameter $d$ is not generally inscribable in a sphere of diameter $d$. To illustrate, your 110-vertex polyhedron actually contains 70 points outside of the diameter-$1$ sphere centered at the origin (I have not verified whether some other diameter-$1$ sphere not centered at the origin might contain fewer than 70 vertices of your polyhedron in its exterior).
With that said, the question of the largest $n$-vertex polyhedron which can be inscribed in a sphere of a given size is asked here. Its solution is unknown for $n>8$, which hints that your problem might be hard to solve.
The accepted answer to that question links to a 1994 page entitled "Maximal Volume Spherical Codes", which appears to use Thomson problem solutions for their putatively optimal arrangements, as they list $n=121$ as the smallest $n$ which breaches the $1/2$ threshold.