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I tried to search similar answers to my problem here, but unfortunately I'm a little bit lost on this subject, originally from a physical problem, but can be stated as:

Let $A\subset R^3$ be a compact and convex set. Let $f: A\rightarrow R^3$ be a nonlinear function defined by $$f(x)=\phi_1(x)+\phi_2(x)$$ where $x\in A$ and $\phi_1$ and $\phi_2$ are nonlinear functions, both being $C^1$. I want to investigate if the image $F\subset R^3$ of $f$ is convex. Physically, $x$ represents the deformation of a section, and $f(x)$ represents the force and moments on it, so obviously $F$ is bounded. I want to know which tools can I apply rather than trying to prove that any segment defined by any two points of $F$ is entirely on $F$.

Luan Gabriel
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  • is $F$ the same as $f$? Why is it obvious that it is bounded? If it is nonlinear, there is no reason to believe it is bounded, unless it is continuous, and hence its image is compact as well – Andres Mejia Mar 22 '16 at 21:18
  • He writes that $F$ is the image of $f$. And the boundedness is due to physical reasons. We can take it as granted. – Friedrich Philipp Mar 22 '16 at 21:20
  • Is $f$ differentiable? Maybe even infinitely many times? – Friedrich Philipp Mar 22 '16 at 21:22
  • Hi @AndresMejia Friedrich, thanks for helping. I will add more background in order to try to clarify the question. So, for each deformation $x$ of the section correspond a image $f$, which means the force and moments on it. The boundness assumption comes from the fact that, on failure of a point in the section, the stress on it can not be an infinite number, i.e., physically the section will break for some $f$. I believe $f$ is differentiable, but I'm still working on it. If it is, what kind of tool can I use to investigate convexity? – Luan Gabriel Mar 23 '16 at 01:54
  • The thing is that while $\phi_2(x)$ is differentiable, I have $\phi_1(x) = [\phi_{11}(x) \phi_{12}(x) \phi_{13}(x)]$ and $$\phi_{11}(x)=\iint_A \sigma(\epsilon) ,du ,dv$$ where $A$ is the area of the section, $\sigma(\epsilon)$ is the nonlinear stress-deformation function, and $$\epsilon (x) = \epsilon_0 + k_yu - k_xv$$ where $x=(\epsilon_0,k_x,k_y)$ and $u,v$ are coordinates of the section. It is given that $\sigma$ is differentiable. The others components of $phi_1$ would follow a similar ideia if $\phi_{11}$ is differentiable. – Luan Gabriel Mar 23 '16 at 02:01
  • I worked on the functions and $f$ is $C^1$. – Luan Gabriel Mar 23 '16 at 21:28

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