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I want to prove or disprove the following:

For any $x_1,y_1,x_2,y_2\in \mathbb{R}$ such that

$$ a \leq x_1 \leq b, \qquad c \leq y_1 \leq d$$ $$ a \leq x_2 \leq b, \qquad c \leq y_2 \leq d$$

when $a,b,c,d$ are constants, and for any $w\in[0,1]$, there exist $x_3,y_3$ such that

$$ a \leq x_3 \leq b, \qquad c \leq y_3 \leq d$$ $$ w(1+x_1)e^{-y_1} + (1-w)(1+x_2)e^{-y_2} = (1+x_3)e^{-y_3} $$

It seems true to me, but I can't prove formally. How can I do this? It is in fact to prove/disprove a set drawn by a function $(1+x)e^{-y}$ is convex or not.

Chang
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1 Answers1

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The map $[a,b]\times [c,d]\to\mathbb R$, $(x,y)\mapsto (1+x)e^{-y}$ is continuous, hence the image of the connected set $[a,b]\times [c,d]$ is connected. Every connected subset of $\mathbb R$ is convex (is in fact an interval).

Hagen von Eitzen
  • 374,180