Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric. Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a point $x\in X$ corresponds to the one-point set $\{x\}\subset X$.

A metric space $X$ will be called nice if there is a short (= 1-lipschitz) retract $\mathrm{Haus}\,X\to X$. In other words, if for any compact set $A\subset X$ there is a point $p_A\in X$ such that $p_{\{x\}}=x$ and $$|p_A-p_B|_X\leqslant |A-B|_{\mathrm{Haus}\,X}.$$

For example, any injective space is nice. Also, discrete spaces (all distances = 1) are nice. On the other hand, Euclidean plane is not nice.

Is there an example of a nice compact geodesic space which is not injective?

Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric. Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a point $x\in X$ corresponds to the one-point set $\{x\}\subset X$.

A metric space $X$ will be called nice if there is a short (= 1-lipschitz) retract $\mathrm{Haus}\,X\to X$. In other words, if for any compact set $A\subset X$ there is a point $p_A\in X$ such that $p_{\{x\}}=x$ and $$|p_A-p_B|_X\leqslant |A-B|_{\mathrm{Haus}\,X}.$$ (Note that we do not assume that $p_A\in A$.)

For example, any injective space is nice. Also, discrete spaces (all distances = 1) are nice. On the other hand, Euclidean plane is not nice.

Is there an example of a nice compact geodesic space which is not injective?

Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric. Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a point $x\in X$ corresponds to the one-point set $\{x\}\subset X$.

A metric space $X$ will be called nice if there is a short (= 1-lipschitz) retract $\mathrm{Haus}\,X\to X$.

For example, any injective space is nice. Also, discrete spaces (all distances = 1) are nice. On the other hand, Euclidean plane is not nice.

Is there an example of a nice compact geodesic space which is not injective?

Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric. Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a point $x\in X$ corresponds to the one-point set $\{x\}\subset X$.

A metric space $X$ will be called nice if there is a short (= 1-lipschitz) retract $\mathrm{Haus}\,X\to X$. In other words, if for any compact set $A\subset X$ there is a point $p_A\in X$ such that $p_{\{x\}}=x$ and $$|p_A-p_B|_X\leqslant |A-B|_{\mathrm{Haus}\,X}.$$

For example, any injective space is nice. Also, discrete spaces (all distances = 1) are nice. On the other hand, Euclidean plane is not nice.

Is there an example of a nice compact geodesic space which is not injective?