$f$-vector inequalities for convex polytopes

During our (joint work with Anand Kulkarni) on polytope enumeration we have accumulated polytope datasets for d=3,4,5,6,7,8,9

The data sets from these experiments are being used to derive new $f$-vector inequalities.

Consider the following example:
1 F_VECTOR ( 10 25 30 20 7 1 )
1 F_VECTOR ( 12 30 34 21 7 1 )
1 F_VECTOR ( 14 35 40 25 8 1 )
2 F_VECTOR ( 16 40 44 26 8 1 )
3 F_VECTOR ( 18 45 48 27 8 1 )
3 F_VECTOR ( 18 45 50 30 9 1 )
2 F_VECTOR ( 20 50 52 28 8 1 )
7 F_VECTOR ( 20 50 54 31 9 1 )
15 F_VECTOR ( 22 55 58 32 9 1 )
7 F_VECTOR ( 22 55 60 35 10 1 )
28 F_VECTOR ( 24 60 62 33 9 1 )
34 F_VECTOR ( 24 60 64 36 10 1 )
25 F_VECTOR ( 26 65 66 34 9 1 )
116 F_VECTOR ( 26 65 68 37 10 1 )
1 F_VECTOR ( 28 70 68 34 9 1 )
1 F_VECTOR ( 28 70 69 34 9 1 )
36 F_VECTOR ( 28 70 70 35 9 1 )
350 F_VECTOR ( 28 70 72 38 10 1 )
1 F_VECTOR ( 28 70 73 39 10 1 )
1 F_VECTOR ( 30 75 72 35 9 1 )
2 F_VECTOR ( 30 75 73 35 9 1 )
19 F_VECTOR ( 30 75 74 36 9 1 )
596 F_VECTOR ( 30 75 76 39 10 1 )
3 F_VECTOR ( 30 75 77 40 10 1 )
9 F_VECTOR ( 32 80 78 39 10 1 )
11 F_VECTOR ( 32 80 79 39 10 1 )
1252 F_VECTOR ( 32 80 80 40 10 1 )
7 F_VECTOR ( 32 80 81 41 10 1 )
18 F_VECTOR ( 34 85 82 40 10 1 )
39 F_VECTOR ( 34 85 83 40 10 1 )
2 F_VECTOR ( 34 85 83 41 10 1 )
1575 F_VECTOR ( 34 85 84 41 10 1 )
22 F_VECTOR ( 34 85 85 42 10 1 )
16 F_VECTOR ( 36 90 86 41 10 1 )
37 F_VECTOR ( 36 90 87 41 10 1 )
7 F_VECTOR ( 36 90 87 42 10 1 )
1212 F_VECTOR ( 36 90 88 42 10 1 )
42 F_VECTOR ( 36 90 89 43 10 1 )
60 F_VECTOR ( 38 95 90 42 10 1 )
75 F_VECTOR ( 38 95 91 42 10 1 )
16 F_VECTOR ( 38 95 91 43 10 1 )
1170 F_VECTOR ( 38 95 92 43 10 1 )
43 F_VECTOR ( 38 95 93 44 10 1 )
1 F_VECTOR ( 40 100 93 42 10 1 )
61 F_VECTOR ( 40 100 94 43 10 1 )
162 F_VECTOR ( 40 100 95 43 10 1 )
10 F_VECTOR ( 40 100 95 44 10 1 )
756 F_VECTOR ( 40 100 96 44 10 1 )
1 F_VECTOR ( 42 105 100 44 10 1 )
203 F_VECTOR ( 42 105 100 45 10 1 )
4 F_VECTOR ( 42 105 97 43 10 1 )
45 F_VECTOR ( 42 105 98 44 10 1 )
183 F_VECTOR ( 42 105 99 44 10 1 )
2 F_VECTOR ( 44 110 100 44 10 1 )
6 F_VECTOR ( 44 110 101 44 10 1 )
34 F_VECTOR ( 44 110 103 45 10 1 )
1 F_VECTOR ( 6 15 20 15 6 1 )


Analyzing this set of 5d polytopes we can get many linear inequalities, the hope is to get new ones which are hitherto unknown, something like

6f_3 + 2f_2 < 8(f_0+f_4)