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the language $\{1011\}$ can't be pumped because there is no way we can apply the pumping lemma on it and get something in the language. What would be the pumping length?

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Pumping length 5. There aren't any strings of length greater than or equal to 5 in $\{1011\}$, so the condition for the pumping lemma is vacuously true.

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    What about the minimum pumping length for the language of the empty string?. Is it 1? – 2017-02-19
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    Yes, it is for the same reason. – 2017-02-19
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    and the language of $\sum^*$ 1 as well right? – 2017-02-19
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    Yes, but for a different reason. – 2017-02-19
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    Considering all the possibilities for $\sum*$. It's either empty or infinite. If it's empty, then it just contains the empty string. If it's infinite then it's alphabet at least contains one character. Both cases result in a minimum pumping length of 1. Is that correct? – 2017-02-19