This nice article
Megan Martin, Cornelia A. Van Cott & Qiyu Zhang (2024) The Beauty of Halving it All, Math Horizons, 31:2, 14-17, DOI: 10.1080/10724117.2023.2249357.
shows that the envelope of lines that partition a triangle into two equal area regions (halving lines) consists of three hyperbola arcs.
My question is: What are the surfaces that form the envelope of halving planes of a tetrahedron? I'd be interested even in just an image, if the analytic calculations are too complicated.
Answered by Jean Marie:
Beyer, W. A., and Blair Swartz. "Bisectors of triangles and tetrahedra." The American Mathematical Monthly 100, no. 7 (1993): 626-640.
"The envelope $E$ also consists of seven pieces-it is like a heptahedron whose faces have been pinched to tangency along its edges..."
