Let $X$ be a closed convex set in $\mathbb R^n$ and $y\in \mathbb R^n$. Suppose a projection $T_X: \mathbb R^n \to \mathbb R^n$ satisfies
$$T_X( \alpha y+ (1-\alpha) T_X(y))=T_X(y)$$ for all $\alpha\in[0,1]$ and $$T_X(y)=y$$ if $y\in X$.
Then what would be implications of the above conditions?
Is it possible to show that $T_X$ must be of the form
$$P_X(y)=\arg\inf_{x\in X}\|x- y\|_a$$
where $\|.\|_a$ is from some parameterized family, i.e. $1\leq a\leq \infty$ ?
Thanks!
The equation means, that the projection of points on the line segment between $y$ and $T_X(y)$ is $T_X(y)$. All projections in Hilbert space onto a non-empty, closed, and convex set share this property.
To see this, let $C$ be non-empty, closed, and convex with the $X\subset C$. Define $$ P_C(y):= \operatorname{arg min}_{x\in C}\|x-y\|_2. $$ Then $P_C$ has the properties mentioned for $T_X$ in your post. Hence, $T_X$ is not necessarily the projection onto $X$, but can be the projection onto any super-set of $X$.
The mapping $T_X$ is not necessarily a projection. Define $X=\{(x_1,x_2): x_1\in[0,1], x_2 =0\}\subset \mathbb R^2$. Define $T_X$ by $$ T_X(x) = \begin{cases} x & \text{ if } x\in X,\\ (0,0) & \text{ if } x_1<1/2, x_2 \ne0\\ (1,0) & \text{ if } x_1\ge1/2, x_2 \ne0. \end{cases} $$ This should satisfy that the line between $y$ and $T_X(y)$ is mapped to $T_X(y)$. Moreover, $T_X$ is discontinuous, so it cannot be a projection onto a convex set in a Hilbert space.