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Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By taking $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exist 3 such points.

This is an interesting problem in differential geometry.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By taking $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exist 3 such points.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By taking $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exist 3 such points.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By taking $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exist 3 such points.

This is an interesting problem in differential geometry.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By takin $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exists 3 such points.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

By taking $P$ to be the plane passing through maximum and minimum points of $u$ we can trivially get two points $x_1,x_2$ with $\nabla u(x_i) \cdot n =0$, $i=1,2$. I wonder if there exist 3 such points.