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Let $C\subset \mathbb{R}^n$ a convex set that has nonempty interior.

Show: $y\in\mathrm{cl}C,\;x\in\mathrm{Int}C \; \implies\; \lambda x+(1-\lambda)y\in\mathrm{Int}C$

My attempt: The case $y\in\mathrm{Int}C$ is obvious, I am struggling with $y\in\partial C$, so $y$ on the boundary of $C$.

stack_math
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  • see, e.g., https://math.stackexchange.com/questions/7376/why-does-a-convex-set-have-the-same-interior-points-as-its-closure/20473#20473 – daw May 16 '22 at 07:10
  • The answer sadly leaves my case as an exercise, so no. The other answers just prove $\dot{\overline{C}}=\dot{C}$, which would only recharacterize my problem. – stack_math May 16 '22 at 07:42
  • well. what about https://math.stackexchange.com/questions/3503970/is-the-convex-combination-of-an-interior-point-and-a-boundary-point-also-in-the?rq=1 – daw May 16 '22 at 07:54
  • This assumes that $C$ is closed.. – stack_math May 16 '22 at 08:01
  • This should give you an idea how to argue for $y\in C$, then generalize to $y \in cl \ C $. – daw May 16 '22 at 08:17

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