We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{ R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{ 1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{ dim}(\{ y\in \mathbb{ R}^n \setminus A\mid \text{ dim} (\pi_y(A)) < s\})\leq \max\{ k+s -\dim A,0\}. \] The second conjecture is by Liu: Given a Borel set $A\subset \mathbb{ R}^n$, then \[ \text{ dim} (\{ x\in \mathbb{ R}^n \setminus A \mid \text{ dim}(\pi_x(A))<\text{ dim} A\}) \leq \lceil \text{ dim} A\rceil. \]
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