Sufficient conditions for connectedness of indifference sets of a preference relation defined on a compact and convex set only

Question

Let $\succsim$ a complete, reflexive and transitive binary relation defined on $X$, a non-degenerated (i.e not identical to a singleton) convex compact subset of $\mathbb{R}^n_{++}$ (the set of n-dimensional vectors with positive components). Suppose $\succsim$ is continuous for the order topology and strictly monotonic (i.e, it preserves the usual partial order of $\mathbb{R}^n_{++}$).

Are the associated indifference sets connected ?

For a preference relation defined on the whole convex cone $\mathbb{R}^n_{++}$ the properties I introduced are indeed sufficient, as shown in this post Are Indifference Curve graphs continuous given the preferential definition of continuity?

Can you think of a counter-example or, on the contrary, provide the proof of the sufficiency of these conditions ?

Thank you very much !

Answer 1

No. Actually, standard preferences restricted to a budget line will not have this property.

For concreteness, take the preferences on $\mathbb{R}^2_{+}$ defined by the utility function given by $u(x_1,x_2)=\sqrt{x_1}+\sqrt{x_2}$. Clearly, these preferences have all the desired properties. Now let $$X=\{(x_1,x_2)\in \mathbb{R}^2_{+}\mid x_1+x_2=1\}.$$ Then the indifference curve corresponding to $(1,0)$ when restricted to $X$ is $\{(1,0),(0,1)\}$. Indeed, the preferences are strictly convex, and any other point on the line must be, as a proper convex combination, strictly better and, therefore, not indifferent.