Is inequality increasing? Does it correspond to increasing social harm? I have a series of related questions. 

My understanding is. 


Economic wealth (ie. GDP) per capita is increasing around the world. We can refer to this as 'the pie is getting bigger'. 
At the same time, inequality (the distribution of the pie) is also increasing. 
Economic inequality is correlated with social harm, such as crime. 


The questions I have is:


Is it correct to say that inequality is correalated with social harm? What do we mean by this? 
Is inequality increasing consistantly around the world? Or is it decreasing in some places?
If it is increasing, are we also seeing a corresponding increase in social harm?
Regarding your first question:

Usually public economists assume that utility is a concave function of income. Therefore, the utility gain from one additional dollar of income is lower for a rich person than for a poor person. This is consistent with risk averse behavior of individuals which we often observe when individuals face risky choices.

If utility is concave in income and the total income is fixed, then the optimal utilitarian distribution would be that all individuals receive the same amount of income. Any deviation from this optimal income generates some loss in social welfare, which you may call "social harm". However, in many cases the optimal utilitarian distribution of income will still be unequal if a higher level of total income can be reached by allowing for inequality. Reasons for this may be asymmetric information or labor market incentives. Thus, higher inequality alone does not necessarily mean lower social welfare. But higher inequality with lower or equal income means lower social welfare in a utilitarian framework.

Disclaimer: notice that if your definition of social welfare is not utilitarian, none of this applies.

Extended reading: in case you are interested in this subject, I recommend you to read an introductory textbook on public finance or public economics.

For a more detailed survey of how social welfare and inequality are related, see:

Lambert, Peter J. "Evaluating impact effects of tax reforms." Journal of Economic Surveys 7.3 (1993): 205-242.