In the French Toulon woman with a knife shouting "Allahu Akbar!" attacked the visitors of the supermarket.
A customer in the store in the maritime town of La Seyne-sur-Mer was injured in the chest and hospitalized.
The most serious incident of this kind involving box cutters was of course the September 11th attack in 2001, which saw four groups of migrants hijack domestic airliners in the United States and crash three into the World Trade Center and Pentagon building, killing nearly 3,000 people and injuring over 6,000.
Marchal added that the attacker has been detained. The victims were taken to hospital, with their injuries described as non-life-threatening.
The 24-year-old assailant, who did not have a police record, was overpowered by others in the supermarket and has been taken into custody.
A woman working at a checkout counter was also wounded, though not as seriously.
He added that "contrary to what was said at the start, she was not wearing a veil, but a black tunic over black pants and black glasses".
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Regional newspaper Nice-Matin, which first reported the 10:30 a.m. attack, quoted an unidentified witness as saying that people in the store stopped the woman from cutting anyone else.
"I asked her 'Why did you do it.' She did not answer".
A search of the woman's home was underway Sunday afternoon, to know whether "she has links or not with Daesh", said BFMTV.
On November 13, 2015, 130 people were killed in simultaneous coordinated attacks in Paris.
A man also attacked people near the Paris Opera House in May, killing one person and injuring four, an action claimed by the Islamic State.
In March, a Franco-Moroccan man shot dead three people, including a policeman, at a supermarket in the southern town of Trebes before being killed by police.