For the sponge base, beat the butter with the sugar until fluffy, add the eggs one by one and continue beating on high speed. Fold in the flour and pour into a square baking pan.
Bake at 170 °C hot air for about 15 minutes, remove from the oven, brush with the jam and let cool.
Squeeze juice from 3 grapefruits (should yield about 250-300 ml). Bring the grapefruit juice to a boil with the sugar and agar agar powder and simmer for 1-2 minutes, stirring constantly, then stir into the yogurt.
Spread the yogurt-grapefruit mixture on the cooled sponge base.
Cut fillets from the remaining 2 grapefruits, place on top and refrigerate until firm, about 2 hours.
Portion the slices into small pieces and garnish with some mint.