The mid-arch portion of a shoe is called a waist. In general, the waist of a shoe should hourglass only minimally. The following image shows a Puma shoe with a waist that is ridiculously narrow – even for an average width foot.
If you excessively pronate, or have any conditions associated with excessive pronation (plantar fasciitis, posterior tibial tendonitis, Achilles tendonitis), then wearing a shoe lacking support in the waist (midfoot) could prevent you from getting better.
Next time you find yourself shoe shopping, in addition to checking the price tag on the sole of the shoe, also check the width of the waist. Having a wide waist is good for shoes, bad for cardiovascular health.