The space of triangles that fit inside a given triangle, parametrized by edge lengths Given a triangle T with sides a, b, and c, describe its "fitting set," the set of all points (x,y,z) in 3-dimensions for which a triangle with sides x, y, z exists that fits in T.
Such a set lies in the positive octant, is star-shaped with respect to the origin, and probably is a polyhedron; but it seems difficult to describe.  
 A: Here's the abstract of K.A. Post, "Triangle in a triangle: On a problem of Steinhaus", Geom Dedicata (1993) 45: 115; this paper was cited in the one given in the comment by Nemo.

A necessary and sufficient condition on the sides $p, q, r$ of a triangle $PQR$ and the sides $a, b, c$ of a triangle $ABC$ in order that $ABC$ contains a congruent copy of $PQR$ is the following: At least one of the 18 inequalities obtained by cyclic permutation of $\{a, b, c\}$ and arbitrary permutation of $\{p, q, r\}$ in the formula:
  \begin{array}{l} Max\{ F(q^2  + r^2  - p^2 ), F'(b^2  + c^2  - a^2 )\}  \\           +  Max\{ F(p^2  + r^2  - q^2 ), F'(a^2  + c^2  - b^2 )\}  \le 2Fcr \\  \end{array}
  is satisfied. In this formula $F$ and $F'$ denote the surface areas of the triangles, i.e.
  \begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2  + 2b^2 c^2  + 2c^2 a^2  - a^4  - b^4  - c^4 )^{1/2}  \\  F' = {\textstyle{1 \over 4}}(2p^2 q^2  + 2q^2 r^2  + 2r^2 p^2  - p^4  - q^4  - r^4 )^{1/2} . \\  \end{array}

Here's a picture for the case $a=b=c=1$, which I plotted in this SageMath Jupyter notebook.  Note that in this case we need only consider 3 of the above inequalities (plus the triangle inequalities):

