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Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $(X,d)$ of radius $r$ centered at $p$ is given by a $k$-dimensional subset of $(X,d)$: $S^k = \{x\in X|d(p,x) = r\}$. Here dimension I'm referring to covering dimension.

Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $(X,d)$ of radius $r$ centered at $p$ is given by a $k$-dimensional subset of $(X,d)$: $S^k \subset \{x\in X|d(p,x) = r\}$. Here dimension I'm referring to covering dimension.

Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $X$ of radius $r$ centered at $p$ means that the covering dimension of $S^k$ is $k$ and $S^k = \{x\in X|d(p,x) = r\}$.

Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $(X,d)$ of radius $r$ centered at $p$ is given by a $k$-dimensional subset of $(X,d)$: $S^k = \{x\in X|d(p,x) = r\}$. Here dimension I'm referring to covering dimension.

Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. Is the intersection $S \cap S'$ always an $(n-2)$-metric sphere? A $k$-dimensional metric sphere $S^k$ in $X$ of radius $r$ centered at $p$ means that the covering dimension of $S^k$ is $k$ and $S^k = \{x\in X|d(p,x) = r\}$.

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $X$ of radius $r$ centered at $p$ means that the covering dimension of $S^k$ is $k$ and $S^k = \{x\in X|d(p,x) = r\}$.

Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?