Skip to main content
MathOverflow

Return to Question

define normal; fix title typo
Source Link
Alexander Pruss
Alexander Pruss
  • 2.6k
  • 14
  • 28

If a compact convex set meets the positive orthant does it metmeet it at a point with a normal in the positive orthant?

Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?

Here, a vector $v$ is normal to a convex set $C$ at $x$ iff for all $y\in C$, $\langle v,y \rangle\le \langle v,x\rangle$.

If a compact convex set meets the positive orthant does it met it at a point with a normal in the positive orthant?

Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?

If a compact convex set meets the positive orthant does it meet it at a point with a normal in the positive orthant?

Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?

Here, a vector $v$ is normal to a convex set $C$ at $x$ iff for all $y\in C$, $\langle v,y \rangle\le \langle v,x\rangle$.

Source Link
Alexander Pruss
Alexander Pruss
  • 2.6k
  • 14
  • 28

If a compact convex set meets the positive orthant does it met it at a point with a normal in the positive orthant?

Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?