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It appears as though all convex sets are conic. But is every conic set convex? I can't seem to think of an edge case where a conic set isn't convex. Moreover, from the definition of a cone and conic combination:

From the definition of a cone:
Let $\theta \in R$ and $x \in R^{n}$, $x \in C \implies \theta x \in C$.

From the definition of a conic combination:
Let $\theta_{1}, \theta_{2}, ..., \theta_{n} \in R$ and $x_{1}, x_{2}, ..., x_{n} \in R^{n}$, then the linear combination $$\tilde{x} = \theta_{1}x_{1} + \theta_{2}x_{2} + ... + \theta_{n}x_{n}$$ is a cone if $\theta_{1}, \theta_{2}, ..., \theta_{n} \geq 0$.

Don't these definitions encompass convex sets and convex hulls (restricting $\sum_{i = 1}^{n}\theta_{i} = 1$)?

David
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    take the convex interval $[0,1]$, is every multiple of its elements contained in it? The convex combinations are a subset of the conic combinations, but not the other way around. – Fede Poncio Jul 26 '21 at 22:42
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    I think you mixed up your question a little. It is clear from your definition that cones are automatically convex, but convex sets may not be cones. If you want a counterexample of the previous one, take any bounded non-trivial convex set in a non-trivial space. – Theo Bendit Jul 26 '21 at 22:42
  • It certainly is not true that all convex sets are cones (e.g., consider the convex subset ${1}$ of $\Bbb{R}$). Your attempts to give the definition of a cone are not right. See https://en.wikipedia.org/wiki/Conical_combination for the correct definitions. It drops out fairly easily from the correct definition of a cone that cones are convex. – Rob Arthan Jul 26 '21 at 22:42
  • It should also be pointed out that cones are not always defined in this way. Frequently they are not defined to be convex, and instead defined to be sets stable under non-negative scaling. That is, $C$ is defined to be a cone if for every $x \in C$ and $\alpha \ge 0$, we have $\alpha x \in C$. Equivalently, $C$ is just a union of rays from the origin. The property you've described would be equivalent to being convex and being a cone. The (not-necessarily-convex) conical hull of a set $C$ would simply be ${\alpha x : \alpha \ge 0, x \in C}$. – Theo Bendit Jul 26 '21 at 23:04

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