If wealth inequality was lowered
You say this as if it's a thing that he government can just decide to do.
And in any case, the vast majority of homicide per capita is committed by black americans. Race is a stronger predictor of homicide rate than wealth. That is, blacks from higher income families have a higher homicide rate than whites from lower income families.
The problem is, politicians aren't Einstein's by any stretch. Looking at crime specifically, the right wing answer is "lock them up" and impose harsh sentences. As a society, you want to keep the prison population low because incarceration is a huge cost to the taxpayer.
The cost of crime, directly and indirectly, vastly outweighs the cost of incarceration. If you take a soft on crime approach, the damage from increased crime increases far more than what you could hope to save with a small inmate population.
The US in particular is headed towards a silent disaster because they have a prison population full of people doing astronomical sentences. When those prisoners age and aren't due for release, the incarceration costs are going to skyrocket. On the flip side, the left wing answer is more lenient bail and non prison alternatives.
What's an "astronomical sentence" and how many people are seving one?
What really would help crime (beyond the economic issues Ive mentioned), is investing heavily in child protection (i.e. early intervention) and having robust drug, alcohol and mental health treatment options. All of these (which wont win elections) require large upfront costs but long term, they lead to considerable savings and better outcomes for society.
This can only help if you can force people into mental health or rehab facilities without their consent. The left categorically opposes this, so it won't work, end of story.
To quickly digress, most public servants in the know will tell you the reason Queensland has a Youth Crime problem is because Queensland has the WORST child protection system in Australia and it has been that way for a while. But you will never hear Labor or Liberal highlight that...
Queensland's 'youth crime' problem is really an 'indigenous youth crime' problem. Everything, including trying to fix it, is downstream of this fact.