I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the question at math.stackexchange which did not attract much attention there. Following is a particular version of the question.

Let $p>1$ and let

$$L = \{ \ \textbf{x} \in \mathbb{R}^n:\ \ x _i\ge 0, \sum_i x_i=1, \sum_i a_i x_i =b\ \}$$ for some real numbers $a_i$ and $b$.

Suppose that $\textbf{x}, \textbf{y}\in L$ with the following property: $x_i=0$ wherever $y_i=0$, and $y_j\neq 0$ but $x_j=0$ for some $j$, i.e., $\bf{y}$ has more support than $\bf{x}$. Then $\|\textbf{x}\|_p>\|\textbf{y}\|_p$ or there exists $\textbf{z}\in L$ such that $\textbf{z}$ has more support than $\textbf{x}$, i.e., $x_i=0$ wherever $z_i=0$ and $z_k\neq 0$ but $x_k=0$ for some $k$ and $\|\textbf{x}\|_p>\|\textbf{z}\|_p$.

The claim is roughly that the $\ell_p$-norm of points of $L$ which are more closer to the boundary of the probability simplex is larger than the ones which are interior enough. In 2-dimension, the claim is very obvious. The only possibility of $L$ there is $\{ \(x_1,x_2):\ x_i\ge 0, x_1+x_2=1\ \}$ and the boundary points $(1,0),(0,1)$ have the maximum $\ell_p$-norm and as we go deeper interior the norm is lesser and lesser. I am wondering whether the same would be the case even in the higher dimensions.