Packed with antioxidant compounds, pomegranates have long been linked to both heart and brain health. Newer research explains why: One study found pomegranate polyphenols help your arteries expand and contract to manage blood flow and prevent hardening. A separate study found the same antioxidants help ward off the type of inflammation that leads to Alzheimer’s disease. To get the most benefit, eat the fruit’s seeds and some of the pith. Both contain healthful compounds.