I want to calculate the distance between a point $y\in \mathbb{R}^N$ to the hyperplane defined by $a^Tx=0$. Here, the distance metric is the general distance, i.e., $d(x,y)=|| y-x ||_p$, where $p>0$.
The minimum distance problem is equivalent to solve the following problem \begin{align} \min_x || y-x ||_p, ~~\text{s.t.} ~ a^Tx=0, \end{align} where $x,y,a\in \mathbb{R}^N$.
I read some references, it said that the problem above has the minimum distance of \begin{align} \frac{|a^Ty|}{||a||_q} \end{align} where $1/p+1/q=1$. The case for $p=2$ can be found and proved easily in the textbook, but how can we prove the general case?
As Anne Bauval commented, I notice there is a question Distance between point and linear Space. It has the conclusion for arbitrary kernal $f$, and in my case $f(x) = a^Tx$. Maybe we can have some ideas from that question.
Here, I explicitly write the type of norm to avoid confusions. Based on a previous question: Distance between point and linear Space, the minimum distance is given by \begin{align} \frac{|a^T y|} {|| f ||_{op}} \end{align} where \begin{align} || f ||_{op} = \sup_x \frac{| f(x)|}{\| x\|_p} \end{align} In my case, $|| f ||_{op}=\| a\|_q$.
