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Mar 30, 2022 at 18:54 comment added ARG It turns out that the "biaugmented antiprisms"/"gyroelongated dipyramids" are described combinatorially in a paper by David Eppstein (also on the arxiv) and attributed to Michael Goldberg. Although I did not find the mention that there is a member in those families which have an insphere touching the faces at the same point. The "gyrate dipyramid" in the other answer are also in D.Eppstein's paper and are called "biarc hull" there.
Nov 29, 2020 at 19:56 comment added ARG I called these solids "biaugmented antiprisms", but then noticed they could also be called "gyroelongated dipyramids". Note that the Johnson solid called "gyroelongated square dipyramid" is essentially identical to the solid $k=4$ above, but because its faces are equilateral triangles, it does not have an insphere.
Nov 29, 2020 at 0:27 history edited ARG CC BY-SA 4.0
corrected sentence
Nov 28, 2020 at 19:43 history edited ARG CC BY-SA 4.0
corrected case k=3, added image and details of other cases
S Nov 27, 2020 at 19:47 history answered ARG CC BY-SA 4.0
S Nov 27, 2020 at 19:47 history made wiki Post Made Community Wiki by ARG