It is known1 that any convex body $K$ in the plane can be partitioned into $6$ equal-area pieces by $3$ concurrent lines which meet at a point in $K$. Call this a $6$-partition. This result cannot be extended to $8$-partitions (by $4$ concurrent lines): A triangle cannot be $8$-partitioned (again, by $4$ concurrent lines meeting at a point inside the triangle).2
My question is:
Q. Which convex bodies $K$ admit an $8$-partition? Are there natural necessary geometric constraints on $K$? Ideally, a characterization of such $K$?
Although an $8$-partition seems to require a quite symmetric $K$, one can form a non-centrally symmetric $K$ that can nevertheless can be $8$-partitioned, by arranging "bump-outs" on a regular $n$-gon to break symmetries:
An $8$-partition of a convex polygon with no evident symmetries. Each octant contains one "bump-out."
Enlargement of each "Bump-out" above.
Of course the natural generalization to Q is: What can be said about $k$-partitions, $k>6$?
1 Buck, Robert C., and Ellen F. Buck. "Equipartition of convex sets." Mathematics Magazine 22, no. 4 (1949): 195-198. Cited in: Berele, Allan, and Stefan Catoiu. "Nonuniqueness of sixpartite points." American Mathematical Monthly 125, no. 7 (2018): 638-642.
2 Guardia, Roser, and Ferran Hurtado. "On the Equipartitions of Convex Bodies and Convex Polygons." In EuroCG, pp. 47-50. 2000.