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- $\begingroup$ Quite skillful--thanks! More a fitting--rather than an exact analysis as that of Daniel Huber in his answer to the circumscribed case. Puzzled by the 0.11037 volume reported. The convex set has (Euclidean/flat) volume (as indicated at end of question) 1/576 (8 - 6 Sqrt[2] - 9 Sqrt[2] [Pi] + 24 Sqrt[2] ArcCos[1/3]) = 0.00227243 and the circumscribed ellipsoid of DH has volume 1/32 Sqrt[1/ (29 - 12 Sqrt[2])] [Pi] = 0.0283059 (reported in my comment to answer of DH). $\endgroup$Paul B. Slater– Paul B. Slater2020-11-20 13:02:57 +00:00Commented Nov 20, 2020 at 13:02
- $\begingroup$ Thanks for that. I see I made a mistake with the volume calculation. Volume of ellipsoid is pi/6(abc) of the semi-axes and not the sum. I updated my code above to reflect this and obtain as the volume, 0.000179. Would like to know if anyone gets a different value. The calculations (and the coding) are very new to me. $\endgroup$Dominic– Dominic2020-11-20 13:43:33 +00:00Commented Nov 20, 2020 at 13:43
- $\begingroup$ In a comment to his answer, Daniel Huber gives 4 Pi a b c/3 as the volume formula. (I thought I had posted a comment to this effect yesterday, but apparently not.) $\endgroup$Paul B. Slater– Paul B. Slater2020-11-21 15:12:05 +00:00Commented Nov 21, 2020 at 15:12
- $\begingroup$ Thanks. Apparently I don't know the difference between principal and semi-major axes. Volume is 4/3 Pi (abc) where a,b,c are semi-major axes which are specified nicely as the norms of column vectors of the matrix C above. Will update above. This gives volume ratio ellipsoid/convex set of 0.00144/0.00227. Would be nice though to have another way to compute it to compare the answer. Haven't worked with convex optimization before. $\endgroup$Dominic– Dominic2020-11-21 16:57:50 +00:00Commented Nov 21, 2020 at 16:57
- $\begingroup$ Good--glad you're enjoying this whole endeavor. Thanks again for your interest. $\endgroup$Paul B. Slater– Paul B. Slater2020-11-21 23:44:10 +00:00Commented Nov 21, 2020 at 23:44
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