Let me start with an example, let
$$\mathbf{T}=\begin{bmatrix}2&1\\1&3\end{bmatrix}$$
and $α=(2,2)$. Abusing notation, write $\mathbf{λ}^{α}=(λ^2,(1-λ)^2)$ for any $0≤λ≤1$. If we compute $T^{\top}\mathbf{λ}^{α}$ for every value of $λ$, we obtain a series of points $(x,y)$ that connect $(1,2)$ and $(3,1)$ through a curve. As long as for all $i$, $α_i>1,$ this curve goes through a point in which $x$ is minimum (call it $minX$) and another in which $y$ is minimum ($minY$). In the example, $minX=(\frac{3}{4},\frac{19}{16})$ and $minY=(\frac{13}{9},\frac{2}{3}),$ that correspond to weights $(\frac{3}{4},\frac{1}{4})$ and $(\frac{1}{3},\frac{2}{3}),$ respectively.
I'm only interested in those $\mathbf{λ}$ that yield points between $minX$ and $minY,$ i.e., points that are the result of $\frac{1}{3}\leqλ\leq\frac{3}{4}$. In these points, the slope of the curve (represented in the $x,y$ plane) is negative and $minX$ and $minY$ are the "frontier" points that separate the part in which the curve is negatively sloped from those where it is not. In fact, the slope of the curve at $minX$ is zero and it approaches $\infty$ as we move closer to $minY.$
Similarly, for $n>2$, I'd be interested in those points sitting in the part of curved surface that is "negatively sloped". In $\Bbb R^3$, I want to exclude those portions of the surface for which $x,y$ and $z$ increase simultaneously. Of course, I can still find values for $minX,$ $minY$ and $minZ,$ but I don't know how to determine the rest of the "frontier" points.
Additional information: If we define $\mathbf{k}=\mathbf{T}^{\top}\mathbf{\barλ}^{α}$ for any $\mathbf{\barλ}=(\barλ_1,\barλ_2,...,\barλ_n))$, with $\sum_{i=1}^n \barλ_i=1$ and $0\leq \barλ_i \leq 1$, the $\mathbf{λ}$ yielding points between $minX$ and $minY$ have an additional property: they are the ones for which the solution to the problem
$$\max \sum_{i=1}^n \lambda_i$$ $$s.t. T^{\top}λ^{α}=\mathbf{k},$$
denoted $\mathbf{λ}^*$, is such that $\sum_i \lambda_i^*=1.$ Observe that $\mathbf{\barλ}$ is part of the feasible set, and therefore $\sum_i \lambda_i^*\geq1.$ What I would like is to characterize the $\mathbf{λ}$ that are in the frontier between those resulting in $\sum_i \lambda_i^*=1$ and in $\sum_i \lambda_i^*>1.$
(I'm not sure about the tag, so I apologize if I've misled you and I'd be happy to change it if you can suggest a better one).