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The Amplitwist
edit approved Jul 30, 2022 at 9:23
The Amplitwist
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A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexitySufficient criteria of convexity, J. Math. Sci. 10 (1978)., 395–435. Incidentally, this a (hard) exercise in my book my bookLectures on Discrete and Polyhedral Geometry (Exc. 1.25).

A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435. Incidentally, this a (hard) exercise in my book (Exc. 1.25).

A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978), 395–435. Incidentally, this a (hard) exercise in my book Lectures on Discrete and Polyhedral Geometry (Exc. 1.25).

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Igor Pak
Igor Pak
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A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexitySufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435. Incidentally, this a (hard) exercise in my book (Exc. 1.25).

A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435. Incidentally, this a (hard) exercise in my book (Exc. 1.25).

A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435. Incidentally, this a (hard) exercise in my book (Exc. 1.25).

Source Link
Igor Pak
Igor Pak
  • 17k
  • 2
  • 61
  • 123

A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact set and suppose every intersection of $X$ by a plane is contractible. Then $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions. See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435. Incidentally, this a (hard) exercise in my book (Exc. 1.25).