Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>u_i(x)$. Said that, We call $S_x$ the intersection of the upper levels of $u_1, u_2$ at $x$. Let $k≥2$. Let $x_0$ be a point. Call $$V_{x_0}=\{z\in \mathbb{R}^{k}| x_0 \not\in S_{z}, z\not\in S_{x_0} \}$$ a subset in $\mathbb{R}^{k}$. Let $T$ be a subset composed by the points $x$ such that $x_{0}\in V_{x}$ for any $x_0\in T$. I do not think that such a subset is unique. However take any possible subset $T$ in consideration.
Does not $T$ contain a submanifold in $\mathbb{R}^{k}$ with dimension $k$? If yes, can we improve this bound for the set $T$? Can we say that such a bound is $1$. Any simple proof?
