  The phase diagram of the filled skutterudite CeOs$_4$Sb$_{12}$ has been
mapped in fields $H$ of up to 60 T and temperatures $T$ down to 0.5 K using
resistivity, magnetostriction, and MHz conductivity. The valence transition
separating the semimetallic, low-$H$, low-$T$, $\cal{L}$ phase from the
metallic high-$H$, high-$T$ $\cal{H}$ phase exhibits a very unusual,
wedge-shaped phase boundary, with a non-monotonic gradient alternating between
positive and negative. This is quite different from the text-book "elliptical"
phase boundary usually followed by valence transitions. Analysis of
Shubnikov-de Haas oscillations within the $\cal{H}$ phase reveals an effective
mass that increases as $H$ drops toward the $\cal{H-L}$ phase boundary,
suggesting proximity to a quantum-critical point. The associated magnetic
fluctuations may be responsible for the anomalous $H,T$ dependence of the
valence transition at high $H$, whereas the low$-H$, high$-T$ portion of the
phase boundary may rather be associated with the proximity of CeOs$_4$Sb$_{12}$
to a topological semimetal phase induced by uniaxial stress.
